MAST 10008 Workbook

MAST10008

Accelerated Mathematics 1 2015

Exercises

Department of Mathematics and Statistics

University of Melbourne 2015

This compilation has been made in accordance with the provisions of Part VB of the Copyright Act (1968) for the teaching purposes of the

University of Melbourne. No part of this publication may be reproduced or transmitted by any form, except as permitted under this act.

MAST 10008 Accelerated Mathematics 1 Exercises 2015

Contents

Course Information 2

Lecture by lecture outline 4

Sheet 1: Linear Equations and Matrices 6

Sheet 2: Vectors and Solid Geometry 12

Sheet 3: Mathematical Induction, Proofs and Numbers 16

Sheet 4: Complex Numbers 20

Sheet 5: Vector Spaces 24

Sheet 6: Inner Product Spaces 31

Sheet 7: Linear Transformations 35

Sheet 8: Eigenvalues and Eigenvectors 40

Sheet 9: Functions of Two Variables 44

Answers 1: Linear Equations and Matrices 50

Answers 2: Vectors and Solid Geometry 54

Answers 3: Mathematical Induction, Proofs and Numbers 56

Answers 4: Complex Numbers 58

Answers 5: Vector Spaces 61

Answers 6: Inner Product Spaces 65

Answers 7: Linear Transformations 67

Answers 8: Eigenvalues and Eigenvectors 71

Answers 9: Functions of Two Variables 75

1


Course Information

Lecturer: Paul Norbury Room 170

Prerequisite: A study score of 38 or better in VCE Specialist Mathematics, or equivalent.

Credit exclusions for 2015 Handbook subjects: Students may only gain credit for one of

MAST10007 Linear Algebra

MAST10008 Accelerated Mathematics 1

MAST10013 UMEP Mathematics for High Achieving Students

Contact: 47 lectures; 11 tutorials; 11 laboratory classes.

Overview: This subject develops the concepts of vectors, matrices and the methods of linear algebra andintroduces students to differentiation and integration of functions of two variables. Students will be exposed tomethods of mathematical proof. Little of the material here has been seen at school and the level of understandingrequired represents an advance on previous studies. Underlying concepts developed in lectures will be reinforcedin computer laboratory classes.

Topics covered include systems of linear equations, matrices and determinants, vector geometry, lines andplanes, vector spaces, subspaces, linear independence, bases, dimension, inner products, linear transformations,eigenvalues and eigenvectors, complex eigenvalues and exponentials as well as techniques of proof, partial deriva-tives, chain rule for partial derivatives, directional derivatives, tangent planes, extrema for functions of severalvariables and double integrals.

Assessment: Up to 40 pages of written assignments 15% (due during semester), one 45-minute written com-puter laboratory tests 5% (held during semester), a 3-hour written examination 80% (in the examinationperiod).

Recommended text:

H. Anton and C. Rorres, Elementary Linear Algebra, Applications Version, 11th edition, Wiley, 2013.

Website: The subject website is accessed via the my.unimelb student portal athttp://my.unimelb.edu.au

Enter your university email address and password to access the portal. (Note that the first time that you loginyou will need to select Log in using your central e-mail username and password to set up your university emailaccount and password.)

Select Studies from the top menu bar. Then click on LMS from Learning Tools to access the Learning Manage-ment System. Select MAST10008.

Once in the MAST10008 website you will find information specifically for Accelerated Mathematics 1. Later inthe year solutions to past examination papers will be posted on this site.

The website is an important resource for all MAST10008 Accelerated Mathematics 1 students and you shouldcheck the site regularly for new information. Please ensure (early in the year) that you can log on to the website.

Notes: This subject together with Accelerated Mathematics 2 is equivalent, in content, to the three subjectsMAST10006 Calculus 2, MAST10007 Linear Algebra and MAST20026 Real Analysis with Applications. Stu-dents require access to a computer with the software package Matlab installed, currently in every open-accesscampus laboratory. Students are expected to use the software package Matlab but no programming knowledgeis expected.

2


Subject objectives: Students completing this subject will:

be able to use matrix techniques to represent and solve a system of simultaneous linear equations;

understand the extension of vector concepts to abstract vector spaces of arbitrary finite dimension;

understand linear transformations, their matrix representations and applications;

understand the concept and properties of sequences;

be able to differentiate and integrate functions of two variables;

and will be exposed to standard methods of mathematical proof.

Generic skills: In addition to learning specific skills that will assist students in their future careers in science,they will have the opportunity to develop generic skills that will assist them in any future career path. Theseinclude

problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strate-gies;

analytical skills: the ability to construct and express logical arguments and to work in abstract or generalterms to increase the clarity and efficiency of analysis;

collaborative skills: the ability to work in a team;

time-management skills: the ability to meet regular deadlines while balancing competing commitments;

computer skills: the ability to use an appropriate computing package.

Exercise Sheets: This booklet of exercise sheets contains nine separate parts which are numbered Sheet 1through to Sheet 9. These correspond to the different sections of the course and each comprehensively coversthe material you are expected to know. A few of the problems are marked Hard and Very hard. It is notexpected that every student will complete, or even attempt, these particular problems. They are there for thosewho have completed a substantial part of the easier and more routine problems and want some questions whichare a little different and more challenging.

3


MAST10008 Accelerated Mathematics 2015 - Schedule

Week Topic Anton & Rorres

1

Matrices and linear equations

3/3 Systems of linear equations, augmented matrices, row operations. p.2-10

4/3 Gaussian elimination to row echelon form. Rank. p.11-24

5/3 Matrices, matrix algebra, special matrices. p.25-50

6/3 Matrices for elementary row operations. p.51-54

2

10/3 Matrix inverses, augmented matrices, matrix equations. p.55-57

Determinants, Vectors, Solid Geometry

11/3 Determinants, cofactors. p. 93-99

12/3 Determinants by row operations, properties. p.100-105

13/3 Vectors in R3, dot product. p.130-142

3

17/3 Cross product, triple product and applications. p.161-170

18/3 Solid geometry, equations of lines in Cartesian and vector form. p.152-160

19/3 Equations of planes in Cartesian and vector form. p.152-160

Number systems and method of proof

20/3 Natural numbers N, integers Z, rationals Q, mod 2 Z2. Laws of arithmetic.

4

24/3 Proof by contradiction, infinitely many primes,

2 not rational.

25/3 Mathematical induction; simple examples involving proving identities. Links on LMS page

26/3 Mathematical induction; proving inequalities.

Complex numbers

27/3 Complex numbers C p.713-717

5

31/3 Complex exponential, de Moivres theorem. p.718-719

Vector Spaces

1/4 Vectors in Rn, abstract and geometric vectors, norm and dot product in Rn. p.119-142

2/4 Real vector spaces, examples other than Rn. p.171-178

EASTER BREAK 3/4 - 12/4

6

14/4 Subspaces p.179-182

15/4 Linear combinations, spanning sets. p.183-189

16/4 Linear dependence and independence. p.190-199

17/4 Basis and dimension, coordinates. p.200-216

4


Week Topic Anton & Rorres

7

21/4 Solution spaces, row and column spaces. p.225-236

22/4 Finding bases. p.237-246

Inner product spaces

23/4 Inner products on vector spaces. p.335-340

24/4 Norm and distance. p.341-344

8

28/4 Angles, Cauchy-Schwartz inequality. p.345-351

29/4 Orthonormal bases, Gram-Schmidt process. p.352-365

30/4 Least squares estimation, curve fitting. p.366-380

Linear transformations

1/5 Linear transformations. p.433-438

9

5/5 Image and kernel spaces, rank and nullity. p.438-444

6/5 Matrix representations of linear transformations. p.458-461

7/5 Change of basis. p.217-224

8/5 Matrix representations for general bases p.461-467

10

Eigenvalues, eigenvectors

12/5 Eigenvalues and eigenvectors, characteristic polynomial. p.295-298

13/5 Finding Eigenvectors p.299-304

14/5 Diagonalisation. p.305-309

15/5 Powers of a matrix . p.310-315

11

19/5 Markov chains. p.553-562

20/5 Diagonalisation of symmetric matrices. p.389-404

21/5 Application: conic sections and quadric surfaces. p.405-416

Functions of two variables Calc. 1&2 Hass et al.

22/5 Functions of two variables, sketching surfaces, partial derivatives. p.702-730

12

26/5 Tangent plane to a surface, linear approximation. p.747-755

27/5 Directional derivative, gradient, steepest ascent/descent. p.739-746

28/5 Stationary points for a function of two variables. p.756-765

29/5 Double integrals over rectangular domains. p.785-790

5


Problem Sheet 1: Linear Equations and Matrices

Linear Equations

1. Which of the following systems of equations are linear?

(a) x1 3x2 = x3 4

x4 = 1 x1x1 + x4 + x3 2 = 0

(b) 3x xy = 1

x+ 2xy y = 0

(c) y = x 1

x = 1 y

(d) x2 = y

x+ y = 1

The row reduction algorithm

2. Following the algorithm, reduce the following matrices to row echelon form, and then to reduced rowechelon form. Keep a precise record of the elementary row operations you use.

(a)

4 8 16

1 3 6

2 1 1

(b) 2 + i 2 + i 5 6 + i

1 2i 1 2i 2 + i 2 i

(c)

1 2

1 1

2 2

0 2

(d)

0 2 1 4

0 0 2 6

1 0 3 2

(e)

1 2 0 1

2 4 1 1

3 6 1 1

(f)

0 0 2 7

1 1 1 1

1 1 4 5

2 2 5 4

Homogeneous linear equations

3. For each of the matrices of the preceding problem write down the corresponding homogeneous system, therank of the matrix (where rank is the number of leading entries or non=zero rows in row echelon form)and solve this system of equations from the reduced row echelon form. N.B. Elementary row operationsleave columns with all zeros unchanged.

4. What is the rank (where rank is the number of leading entries or non=zero rows in row echelon form) ofthe n by n matrix with 1 in every position? What is the rank of the chessboard matrix with (i, j) entryai,j satisfying ai,j = 0 when i+ j is even and ai,j = 1 when i+ j is odd.

5. Using row echelon reduction, solve the following homogeneous systems:

(a) (2 + i)x iy + (3 i)z + u = 0

x+ y (1 i)u = 0

(b) 7x+ 7y 16z + 6w = 0

4x y + 3z + 7w = 0

3x+ y 2z + 4w = 0

6


Inhomogeneous linear equations

6. Use row reduction to find the ranks of the coefficient and augmented matrices of the following systems ofequations. Decide whether the system has (a) no solution vector, (b) a unique solution vector, (c) morethan one solution vector (x, y, z). Solve the systems where possible and describe the system geometrically.

(a) 3x 2y + 4z = 3

x y + z = 7

4x 3y + 5z = 1

(b) x+ 2y z = 1

2x+ 7y z = 3

3x 12y + z = 0

(c) 3x 4y + z = 2

5x+ 6y + 10z = 7

8x 10y 9z = 5

(d) 2x 3y + 5z = 10

4x+ 7y 2z = 5

2x 4y + 25z = 31

7. Using row echelon reduction, find the general solution to the following system of equations:

2x1 + x2 + 3x3 + x4 = 3x1 + x2 + x3 x4 = 6x1 x2 + 3x3 + 5x4 = 12

4x1 + x2 + 7x3 + 5x4 = 3

8. Determine the values of k for which the system of linear equations has (i) no solution vector, (ii) a uniquesolution vector, (iii) more than one solution vector (x, y, z):

(a) kx+ y + z = 1

x+ ky + z = 1

x+ y + kz = 1

(b) 2x+ (k 4)y + (3 k)z = 1

2y + (k 3)z = 2

x 2y + z = 1

(c) x+ 2y + kz = 1

2x+ ky + 8z = 3

(d) x 3z = 3

2x+ ky z = 2

x+ 2y + kz = 1

For the cases (i) and (ii) find the solutions.

9. Determine the conditions on a, b, c so that the system has a solution:

(a) x+ 2y 3z = a

3x y + 2z = b

x 5y + 8z = c

(b) x 2y + 4z = a

2x+ 3y z = b

3x+ y + 2z = c

Find the solutions when they exist.

7


Arithmetic of matrices

10. Evaluate the following matrix products:

(a)

3 4 2

1 3 6

7 1 0

1 2

0 1

1 0

(b) 2 2

1 1

[ 1 2 ]

(c)[

7 + i 6 4 + 3i]

0

1 i

1

(d)

0

1

1

[ 7 6 4 ]

(e)

3 6 0

0 2 2

1 1 1

2

1

1

(f)

4 3

6 6

8 9

12

13

11. Let

A =

1 0 12 3 41 0 2

.Show that A1 = 13 (A

2 2A 4I) where A1 is a matrix such that A1A = AA1 = I.

12. Verify that[cos (1) sin (1)sin (1) cos (1)

] [cos (2) sin (2)sin (2) cos (2)

]=

[cos (1 + 2) sin (1 + 2)sin (1 + 2) cos (1 + 2)

].

13. Write down the 3 by 2 matrices A and B which have entries given by Aij = i + j and Bij = (1)i+j .Calculate ATB.

14. Give 3 by 3 examples of the following:

(a) a diagonal matrix; that is, a matrix A with Aij = 0 if i 6= j.

(b) a scalar matrix; that is, a diagonal matrix A with Aii = Ajj .

(c) a symmetric matrix; that is, a matrix A with Aij = Aji.

(d) an upper triangular matrix; that is, a matrix A with Aij = 0 if i > j.

15. Use trial and error to find 2 by 2 examples of the following:

(a) a non-zero matrix A with A2 = 0;

(b) a matrix B with real entries and with B2 = I;

(c) matrices C and D with no zero entries but with CD = 0;

(d) matrices E and F with EF = FE but EF 6= 0.

8


16. Suppose that a 2 by 2 matrix A satisfies AB = BA for every 2 by 2 matrix B. Set

A =

[a bc d

].

Then AB = BA when B is either of [1 00 0

] [0 10 0

].

What does this say about a, b, c, d? Show that a matrix A which satisfies the above must be a scalarmatrix.

17. Let A be a square matrix satisfying A2 = A and let B be any matrix of the same size. Show that(AB ABA)2 = 0

Inverses

18. Use elementary row operation to find, where possible, the inverses of the following matrices:

(a)

0 0 1

0 1 0

1 0 0

(b)

1 0 0

0 4 3

0 9 7

(c)

4 7 2

3 1 7

2 4 1

(d)

1 2 3

1 7 4

0 9 1

19. Repeat the preceding problem for the complex matrices

(a)

1 + i i1 1 i

(b) 2 i

3 + i 1

20. Find the inverse of the matrix

A =

4 3 2 01 2 3 12 1 1 31 3 1 2

.Hence solve the system

4x1 + 3x2 2x3 = 1x1 + 2x2 3x3 + x4 = 0

2x1 x2 + x3 3x4 = 1x1 3x2 + x3 2x4 = 3

Check your answer by matrix multiplication.

9


Determinants

21. Evaluate the following determinants by reduction to triangular form:

(a)

2 3 1

4 5 2

1 2 3

(b)

4 5 6

1 2 3

0 1 1

(c)

2 1 2

1 2 1

3 0 3

(d)

1 2 3 4

0 1 2 3

0 0 2 1

0 0 3 2

(e)

1 1 2 3

2 1 2 6

1 0 2 3

2 2 0 5

(f)

3 9 27 81

1 1 1 1

2 4 8 16

2 4 8 16

22. Let

|A| =

a b cd e fg h i

= 1 .Find the following determinants:

(a)

a b c

g h i

d e f

(b)a b c

d e f

g h i

(c)d e f

3g 3h 3i

a b c

(d)

2a 2b 2c

2d 2e 2f

2g 2h 2i

(e)

a b c

d+ a e+ b f + c

g 2a h 2b i 2c

23. Using a suitable determinant find

(a) the area of the parallelogram spanned by the vectors (1, 2) and (3, 5),

(b) the volume of the parallelepiped spanned by the vectors (7, 1, 1), (4, 1, 2) and (1, 0,1)

(c) the volume of the parallelepiped spanned by the vectors (1, 1, 1), (4, 1, 2) and (2,1, 0)

What does your answer to (c) tell you about the three vectors?

24. Establish the following factorizations:1 1 1a b ca3 b3 c3

= (b c)(c a)(a b)(a+ b+ c)x y zx2 y2 z2

yz zx xy

= (y z)(z x)(x y)(yz + zx+ xy)

10


25. Evaluate the following determinants:

(a)

1 a bc

1 b ca

1 c ab

(b)

1 a a3

1 b b3

1 c c3

26. The famous Vandermonde determinant is defined by

Dn =

1 1

21 n11

1 2 22 n12

......

......

1 n 2n n1n

Use row operations to evaluate D2, D3, and D4, expressing your answer as a fully factorized expressionto exhibit the fact that Dn = 0 if any two of the parameters 1, 2, . . ., n are equal.

27. Idempotent Matrices A matrix P is called idempotent if P 2 = P . If P is idempotent and P 6= I showthat detP = 0.

11


Problem Sheet 2: Vectors and Solid Geometry

In the following, vectors are denoted by bold type and e1, e2, e3 denote (1, 0, 0), (0, 1, 0), (0, 0, 1) respectively.

Arithmetic of vectors

1. For the given vectors a, b find a + b, 5a 4b, a b, a, and a b.

(a) a = (2, 6, 1), b = (3,3,1)

(b) a = 3e1 4e2 + 21e3, b = e1 + 2e2 5e3

(c) a = e1 + e2, b = e2 + e3

2. Find the following scalar products:

(a) (1, 1, 1) (2, 1,3) (b) (2, 1, 1) (1,3, 7) (c) (

2, , 1) (

2,2, 3)

3. Find the angle between the following pairs of vectors:

(a) (1, 0, 0), (0, 0, 4) (b) (1,1, 0), (0, 1, 1) (c) (2,2, 2), (1, 0, 2)

4. If a = 2e1 +xe2 +e3 and b = 4e1 2e2 2e3, find x such that a is orthogonal to b. Can you find a valueof x so that a and b are in the same direction?

5. Find the following vector products:

(a) (e1 e2 2e3) (2e1 + e2 3e3) (b) (2, 1, 1) (1,3, 7)

6. Find unit vectors orthogonal to the following pairs of vectors:

(a) (2, 0, 1), (3,1, 4) (b) (2, 5, 3), (1,5, 3) (c) (5, 4, 2), (3, 2, 1)

7. Find the area of a triangle with the vertices (1,1, 2), (2, 1, 1), (1, 2, 3). Find a unit vector orthogonalto the plane of this triangle.

8. Find the volume of a parallelepiped with the sides a = (1,2, 1), b = (2, 3,1), c = (4, 2, 3).

Some vector identities

9. If a b = 0 and x + (x b)a = b, find the vector x.

10. Show that a (b c) = c (a b) = b (a c).

11. Prove that a (b c) = (a c)b (a b)c.

Hint. Find ek (b c) for k = 1, 2, 3. Express a (b c) as a sum3k=1 akek (b c).

12. Use Problems 10 and 11 to show that (a b) (c d) = (a c) (b d) (a d) (b c).

Deduce Lagranges identity: a b2 = a2 b2 (a b)2.

12


13. Prove the following vector identities:

(a) a (b c) + b (c a) + c (a b) = 0

(b) (b c) (a d) + (c a) (b d) + (a b) (c d) = 0

Hint. Use Problems 11 and 12.

14. Prove the following properties of the vector product:

(a) a b = b a (b) (a + b) c = a c + b c

(c) a (b + c) = a b + a c (d) a a = 0

Lines in R3

15. Find a parametric equation of the line `:

(a) ` is the line through P (3, 4,7) and parallel to a = (1, 2,3)

(b) ` is the line through the points A(2, 0, 1) and B(3,4, 1).

16. Find the angle between the lines

(a) x 3 = 2 y, z = 1, and x = 3, y + 2 = z 5

(b)x 2

3=y + 5

4=z 1

3and x = y = z .

17. (Hard) Find the distance of the point A from the line `:

(a) A(6, 1, 2), and ` : 2 x = z 2, y = 2

(b) A(3,1,1), and ` is the line through the points P (1, 0,2) and Q(2, 1,1).

18. (Hard) Find the distance between the lines

(a) x 3 = 4y 2 = z + 1 and x = 3 2t, y = t, z = 1 + t

(b)x 3

2= y + 2 =

z

8and

x

3=y + 1

2=z 2

13

19. Find parametric equations of the line of intersection of the planes 2x y z = 0 and x+ y + 5 = 0.

20. (Hard) Find the distance of the point (1, 1, 1) from the line of intersection of the planes x+ y+ z = 0 andx 2y 3 = 0.

21. Find the angle between the line x = 2t 7, y = 4t 6, 3z = t 5 and the plane x = 2s, y = 2t+ s, z =t s

Planes in R3

22. Find the projection of the vector x on to the vector a:

(a) x = (3,4, 5), a = (3, 1, 2)

(b) x = (12,17, 8), a = (1,1, 2).

23. Find a Cartesian equation of the plane

13


(a) through the points (2, 0, 1), (1, 1, 3), (5, 2, 3)

(b) through the point (2, 0, 1) orthogonal to the vector (1, 2, 3).

24. (Hard) Find the distance of the point (1, 1, 1) from the plane 2x+ y + 2z 2 = 0.

25. Find the angle between the planes 2x+ y + 3z = 0 and 3x 2y + 4z 4 = 0.

Miscellaneous problems on lines and planes

26. Find the intersection of the line 2x = 3y = z with the plane x+ 4y + 3z + 14 = 0.

27. Find a Cartesian equation of the plane through the points (2,1, 0) and (5,3, 1) which is parallel tothe line joining the points (3, 5,1) and (0, 3,2). Write down a parametric equation of the line throughthe origin which is orthogonal to this plane, and find where it meets the plane.

28. Find the foot P of the perpendicular from the point A(2, 3, 1) to the plane with equation 2xy+z1 =0. Check by another method that the distance from A to is equal to the distance from A to P .

29. (Exam, 1988) If the line ` is given by the equations 2x y + z = 0, x+ z 1 = 0, and if M is the point(1, 3,2), find a Cartesian equation of the plane

(a) passing through M and `,

(b) passing through M and orthogonal to `.

30. (Hard) The gutters on two roofs meet at right angles and the roofs themselves make angles of and with the horizontal. Show that the line of intersection of the roofs makes an angle with the horizontal of

arcsin

(1

1 + cot2 () + cot2 ()

).

31. (Hard) The legs of a tripod are at right angles to each other. Two legs have the same length a and thethird is longer with length b. The lower ends of the legs are placed on a level floor. What is the height ofthe top of the tripod above the floor?

32. A median of a triangle is the line connecting a vertex to the midpoint of the opposite side. Use vectormethods to show that the three medians of a triangle intersect in a point.

33. An altitude of a triangle is the line through a vertex which is perpendicular to the opposite side. Usevector methods to show that the three altitudes of a triangle intersect in a point.

34. (Hard) Use vector methods to prove Desargues Theorem:

Two triangles ABC and ABC in space are situated so that the lines AA, BB and CC meet in a pointO. Let

P denote the intersection of BC with BC

Q the intersection of CA with C A

R the intersection of AB with AB.

Assuming that all these intersections exist, show that P , Q and R lie on a straight line.

What can be said if, for example, BC is parallel to BC ?

14


35. (Hard) Use vector methods to prove Pappus Theorem:

Points A,B,C lie on a line ` and points A, B, C lie on a line ` which meets `. Let

P denote the intersection of BC and C B

Q denote the intersection of CA and C A

R denote the intersection of AB and AB.

Assuming that all these intersections exist, show that P , Q and R lie on a straight line.

What can be said if, for example, BC is parallel to BC?

36. (Hard) Use vector methods to prove the theorem of Menelaus:

A line cuts the sides AB,BC and CA of a triangle ABC in the points L,M and N , respectively. Provethat

AL

LB BMMC

CNNA

= 1

where the lengths of the segments are given appropriate signs.

15


Problem Sheet 3: Mathematical Induction, Proofs and Numbers

The following notation is adopted in this Problem Sheet:

N = {1, 2, 3, . . .} the set of all natural numbers

Z = {. . . ,3,2,1, 0, 1, 2, 3, . . .} the set of all integers

Numbers

1. Use the definition of sum and product of rational numbers given in the Notes to show that, if a, b, a1, b1, c, dare integers, and if a/b = a1/b1, then

a

b+c

d=a1b1

+c

dand

a

b cd

=a1b1 cd.

2. Use the definition of sum and product of rational numbers given in the Notes to show that, if a, b, c, d, e, fare integers then

a

b ( cd

+e

f) =

a

b cd

+a

b ef

anda

b+ (

c

d+e

f) = (

a

b+c

d) +

e

f.

3. (Hard) Use the usual ordering of the integers to give an ordering of the rational numbers. Show that yourordering does not depend on the particular choice of expression a/b for the rational numbers.

4. (Hard) This is a method of constructing the integers from the natural numbers. It is similar to the waythat the rational numbers are constructed from the integers.Start with all ordered pairs of natural numbers (a, b). Redefine equality for these pairs by defining

(a, b) (c, d)ifandonlyifa+ d = b+ c.

(a) Show that, if (a, b) (c, d) and (c, d) (e, f), then (a, b) (e, f).

Define(a, b) + (c, d) = (a+ c, b+ d).

(b) Show that, if (a, b) (a1, b1) then (a, b) + (c, d) (a1, b1) + (c, d).

(c) Show that, each pair (a, b) can be identified with exactly one of the following types (with c a naturalnumber):

(c+ 1, 1), (1, 1), (1, c+ 1).

(d) Explain how this can be used to give an abstract construction of the integers by identifying (a, b)with a b Z.

5. Show by example

(a) that the sum of two irrational numbers need not be irrational.

(b) Show by example that the product of two irrational numbers need not be irrational.

6. Express the following repeating decimals as rational numbers:

(a) 0.1111 . . . (b) 2.6666 . . . (c) 0.9999 . . .

(d) 0.34999 . . . (e) 0.37373737 . . . (f) 0.001010101 . . .

16


7. (Hard) Show that any decimal fraction which is eventually periodic must represent a rational number.

8. Solve the following equations (mod 2).

(a) x+ 1 = 0(b) x2 + x = 0 (c) x2 + x+ 1 = 0

Proofs

9. Let m denote a natural number. Which of the following equations involving only natural numbers havesolutions that are also natural numbers? If there is a solution, is it unique, in other words, is there onlyone solution?

(a) m+ 1 = 2(b) m+ 2 = 1 (c) 2m = 4

(d) 2m = 3 (e) (0) .m = 42 (f) (0) .m = 0

If we replace natural number by integer, which of the preceding equations, which had no solution inthe natural numbers, now possess solutions in the integers? Which now become solvable if integer isreplaced by rational number?

10. Prove carefully the following results about integers. (You will need to assume some facts about the integersbut you should state clearly what they are and you should make them as simple as possible.)

(a) The square of an even integer is even.

(b) The product of two odd integers is odd.

(c) The sum of two odd integers is even.

(d) The cube of an odd integer is odd.

(e) If k is an odd integer, k2 1 is divisible by 4.

11. Prove that if a1, a2, . . . , an are integers and x is a rational number such that

xn + a1xn1 + a2x

n2 + + an1x+ an = 0,

then x has to be an integer which is a factor of an. This theorem shows in particular that for n N andn > 1, the nth root of any natural number k is irrational, unless k is of the form mn for some naturalnumber m. The theorem can also be used in some cases to deduce that solutions of algebraic equationsare not rational numbers.

12. Prove carefully that if the square of an integer is even then the integer itself is even. (You will need toassume some facts about the integers but you should state clearly what they are and you should makethem as simple as possible.)

13. One of the following statements is true and the other is false. For the one that is true, give a careful proof.For the one that is false, give an example to show that it is false.

(a) If a natural number n is divisible by 12 then its square is also divisible by 12.

(b) If the square n2 of a natural number n is divisible by 12 then n is also divisible by 12.

Describe the largest set S of natural numbers for which the following is true:

if a S then, for any natural number n, n is divisible by a if and only if n2 is divisible by a.

Give a brief argument to justify your claim.

17


14. Prove that there is no rational number whose square is six. (You should imitate the proof for

2 in theNotes.)

15. Prove that

2 +

3 is irrational.

16. (Hard) Prove that log10 2 is irrational.

17. (Hard) Given 5 points in a square of side 1, show that there are two points of the five for which the

distance apart is no more than22 .

18. (Very hard) Suppose that the points of the plane are each coloured either red, yellow or blue. Prove thatthere are two points at distance one apart which have the same colour.

19. (Very hard) Prove that, for any 1993 integers, there is a subset whose sum is divisible by 1993. Summation notation

20. Rewrite the left hand side of each equality in Problem 1 using the summation notation.

21. Derive the following formulae (without using mathematical induction):

(a)

nk=1

(2k 1) = n2 (Hint: 2k 1 = k2 (k 1)2.)

(b)

nk=1

k =1

2n2 +

1

2n (Hint: Use the preceding formula.)

(c)

nk=1

k2 =1

3n3 +

1

2n2 +

1

6n (Hint: k3 (k 1)3 = 3k2 3k + 1.)

(d)

nk=1

1

k(k + 1)= 1 1

n+ 1=

n

n+ 1(Hint:

1

k(k + 1)=

1

k 1k + 1

.)

Set theory and Inequalities

22. Transform each of the following inequalities into an equivalent inequality free of the modulus sign, suchas a < x < b. Simplify as much as possible.

(a) |x| < 3 (b) |x 2| < 5

(c) |3 2x| < 1 (d) |1 + 2x| 6 3

(e) |x+ 2| > 5 (f) |5 x1| < 1

(g) |x 5| < |x+ 1| (h) |x2 2| 6 1

23. Rewrite each of the following inequalities in terms of intervals; (for example, |x| > 1 becomes (,1)(1,+)).

(a) |x+ 3| > 1 (b) |x 2| < 3

(c) |x 2| < 3 or |x+ 1| < 1 (d) |x 2| < 3 and |x+ 1| < 1

(e) |x+ 2| 6 2 and |x| > 1 (f) |x+ 2| 6 2 or |x| > 1

18


Mathematical induction

24. Prove by induction that the given formula is true for every positive integer n.

(a) 2 + 4 + 6 + + 2n = n(n+ 1)

(b) 1 + 4 + 7 + + (3n 2) = 12n(3n 1)

(c) 2 + 7 + 12 + + (5n 3) = 12n(5n 1)

(d) 1 + 2 2 + 3 22 + 4 23 + + n 2n1 = 1 + (n 1)2n

(e) 12 + 22 + 32 + + n2 = 16n(n+ 1)(2n+ 1)

(f)1

1 2+

1

2 3+

1

3 4+ + 1

n(n+ 1)=

n

n+ 1

(g) 3 + 32 + 33 + + 3n = 32 (3n 1)

(h)(1 + 25 + + n5

)+(1 + 27 + + n7

)= 2

{n(n+1)

2

}4(i) 1 + r + r2 + + rn1 = 1 r

n

1 rif r 6= 1

25. Prove by induction that the following statements are true for every positive integer n.

(a) 3 is a factor of n3 n+ 3

(b) 9 is a factor of 10n+1 + 3 10n + 5

(c) 4 is a factor of 5n 1

(d) x y is a factor of xn yn

(e) 72n 48n 1 is divisible by 2304

26. Prove that the following inequalities hold for all n N.

(a) (1 + x)n > 1 + nx if x > 1 (the so called Bernoullis inequality)

(b) 13 + 23 + + (n 1)3 < 14n4 < 13 + 23 + + n3

(c) 1 +12

+13

+ + 1n>n

(d) (Hard) 1 + 12

+ 13

+ . . . 1n6 2n 1

27. In each case find n0 N such that the inequality holds for all n > n0, and give a proof by induction.

(a) n! > 2n(b) n! > 3n2 (c) n! > 2n3

(d) 2n > n2 (e) 2n > 2n3 (f) 22n > 20 3n

19


Problem Sheet 4: Complex Numbers

Arithmetic of complex numbers

1. Calculate the following expressions and give the result in Cartesian form:

(a) (2 + 3i)(4 i) + (1 + 2i) (b) (2 + i)4

(c)2 3i4 i

(d) (4 + i)(3 + 2i)(1 i)

(e) (2i 1)2{

4

1 i+

2 i1 + i

}(f)

i4 + i9 + i16

2 i5 + i10 i15

(g)(2 + i)(3 2i)(1 + 2i)

(1 i)2(h) 3

(1 + i

1 i

)2 2

(1 i1 + i

)32. Find real numbers x and y such that

(a) 3x+ 2iy ix+ 5y = 7 + 5i

(b) (3 + 4i)2 2(x iy) = x+ iy

(c) (3 2i)(x+ iy) = 2(x 2iy) + 2i 1

(d)

(1 + i

1 i

)2+

1

x+ iy= 1 + i

3. Prove the following facts:

(a) z1 + z2 = z1 + z2 (b) z1z2 = z1z2 (c) |z1z2| = |z1| |z2|

(d) |z| = |z| (e) 1/z = 1/z, z 6= 0 (f) |1/z| = 1/|z|, z 6= 0

4. If


7. Sketch the set of points in the complex plane representing the complex numbers z satisfying the followingrelations:

(a) |z| = 2 (b) |z| < 2 (c) |z| > 2

(d) |z 1| = 2 (e) |z + 1| < 1 (f) |z 2| = 2|z + 1|

(g) |z + i| = |z 1| (h) |z a| = |z b|

8. Consider the triangle in the complex plane which has vertices at the origin, at z1 and at z1 + z2. Byconsidering the sides of this triangle show that

|z1 + z2| 6 |z1|+ |z2|.

9. (Hard) Use mathematical induction to show that, for complex numbers z1, . . . , zn,

|z1 + + zn| 6 |z1|+ + |zn|.

10. Show, using complex numbers, that the line joining the midpoints of two sides of a triangle is parallel tothe third side and half its length.

11. (Hard) Consider a polygon in the complex plane which has vertices z1, . . . , zn (considered as we go roundthe polygon clockwise). Consider the expression

A =1

2=(z1z2 + z2z3 + + zn1zn + znz1).

Show that A is unchanged by translations and by rotations of the polygon about the origin. What is thegeometric meaning of A?

Hint. Try this with n = 3 first.

12. (Hard) A median of a triangle is a line joining a vertex to the midpoint of the opposite side. The centroidof the triangle is the intersection of the medians. Let A be the sum of the squares of the distances fromthe vertices to the centroid of a triangle. Let B be the sum of the squares of the lengths of the sides ofthe triangle. Show that

B = 3A.

Polar form and exponential form

13. Express each of the following complex numbers in the polar form rei. Illustrate with a sketch.

(a) 2 + i2

3 (b) 5 + 5i (c)

6 i

2

(d) 3 + 4i (e) 1 i

3 (f) (1 + i

3)2

(g)1 + i

1 i(h) 1+i

3

1i3

(i) (2 + 3i)(1 2i)

14. Express the following numbers in Cartesian form:

(a) e2+i4 (b) ei (c) ei2

21


15. Express in exponential form ez:

(a) 1 i

3 (b) 3 (c) 1 + i

Is exponential form unique?

16. (a) Compute (2 + i)(3 + i) and use the result to prove that

4= arctan

(1

2

)+ arctan

(1

3

).

(b) Compute (5 i)4(1 + i) and use the result to prove that

4= 4 arctan

(1

5

) arctan

(1

239

).

The latter expression has frequently been used to calculate . The first term on the right hand side iseasy to calculate and the second term converges very quickly.

Applications of exponential form

17. Find the fourth derivative with respect to t of eat cos (bt) using that of e(a+ib)t. Show that

d4

dt4(et cos (2t)) = 25et sin(2t+ )

where = arctan(

724

).

18. For z C find a general formula forezt dt, and use it to calculate

3t cos(t) dt.

19. Use the complex exponential to show that

(a)et cos (3t) dt = 110e

t [cos (3t) + 3 sin (3t)] + C

(b)e2 [2 cos () 3 sin ()] d = 15e2[8 sin () cos ()] + C

(c) 0et sin2 (t) dt = 0.4(1 e)

20. Using the complex exponential, calculate

(a)d51

dt51et cos

(3t)

(b)d75

dt75et sin (t)

21. Prove the following identities:

(a) sin(5)/ sin() = 16 cos4() 12 cos2() + 1 under suitable conditions on

(b) sin(4)/ sin() = 2 cos(3) + 2 cos() under suitable conditions on

(c) sin3() = 34 sin()14 sin(3)

(d) cos4() = 18 cos(4) +12 cos(2) +

38

22. Use de Moivres formula to express the following functions as polynomials in sin() and cos(). Which ofthem can be expressed as polynomials in (a) cos() alone, (b) sin() alone?

(a) cos(4) (b) sin(3) (c) cos(5) (d) sin(4)

22


23. Express the following as linear combinations of trigonometric functions of integral multiples of :

(a) cos5() (b) sin5() (c) cos6() (d) cos3() sin2()

24. (Hard) Give an argument to show that, for any natural number n, cos(n) can be expressed in the form

cos(n) = Pn(cos())

where Pn is a polynomial of degree n. (For example, P1(x) = x and P2(x) = 2x2 1.)

(These polynomials are called Chebyshev polynomials.)

Finding roots and solving equations

25. Find all cube roots of unity and locate them graphically.

26. Find the square roots of i.

27. Find all complex numbers z for which z3 = 8i and sketch them in the complex plane.

28. Find the sixth roots of 64 and illustrate with a diagram.

29. Solve the following equations and locate the zeros graphically.

(a) z5 = 32 (b) z3 = 1 + i (c) z4 = 2

3 2i

30. Find all zeros of the following equations:

(a) z4 2z2 + 4 = 0 (b) z6 + 2z3 + 2 = 0

(c) z4 + 4z2 + 16 = 0 (d) z4 + 1 = 0

(e) z2 + (1 + i)z 123

+ i = 0 (f) z3 (1 + 2i)z2 + iz + i = 0

Suggestion. (vi) z = 1 is a zero. Divide by z 1.

23


Problem Sheet 5: Vector Spaces

Vector Spaces, AR5.1

1. Determine whether or not the given set is a vector space under the usual operations. If it is not a vectorspace, list all properties that fail to hold.

(a) The set of all 2 3 matrices whose second column consists of 0s.

(b) The set of all (real) polynomials with positive coefficients.

(c) The set of all (real valued) continuous functions with the property that the function is 0 at everyinteger, for example f(x) = sin(x).

2. (Function spaces) Show that the set F(S,F) of all functions f : S F is a vector space under thepointwise operations

(f + g)(s) = f(s) + g(s), (f)(s) = f(s)

for all s S taking the scalar field F to be R or C.

Note. Most of the real and complex vector spaces encountered in this course are subspaces of Rn,Cn orF(S,F) for some S and F = R or C.

3. Prove the following consequences of the vector space axioms:

(a) 0 = 0; (b) 0x = 0; (c) (1)x = x.Subspaces, AR5.2

4. For each of the following subsets of R2 sketch the set, then determine whether it is (i) closed underaddition, (ii) closed under scalar multiplication, (iii) a subspace of R2.

(a) {(x, y) : y > 0} (b) {(x, y) : x = y}

(c) {(x, y) : x2 + y2 6 1} (d) {(x, y) : xy = 0}.

5. Decide which of the following are subspaces of R3. Explain your answers.

(a) {(a, b, 0) R3 : a, b R} (b) {(a, b, c) R3 : 2a 3b+ 5c = 4}

(c) {(a, b, c) R3 : 2a 3b+ 5c = 0} (d) {(a1, a2, a3) R3 : a1 > 0}

(e) {(a b, a+ b, 2a) R3 : a, b R}

6. Show that the following sets of vectors are subspaces of Rm.

(a) The set of all linear combinations of the vectors (1, 0, 1, 0) and (0, 1, 0, 1) (of R4).

(b) The set of all vectors of the form (a, b, a b, a+ b) (of R4).

(c) The set of all vectors (x, y, z) such that x+ y + z = 0 (of R3).

7. Show that the following sets of vectors are not subspaces of Rm.

(a) The set of all vectors whose first component is 2.

(b) The set of all vectors except the vector 0 .

(c) The set of all vectors the sum of whose components is 1.

24


8. Use the subspace theorem to decide which of the following are real vector spaces with the usual operations.

(a) The set of real polynomials of any degree.

(b) The set of real polynomials of degree 6 n.

(c) The set of real polynomials of degree exactly n.

(d) The set of real polynomials p with p(0) = 0.

(e) The set of real polynomials p with p(0) = 1.

(f) The set of all differentiable functions.

(g) The set of all solutions of the differential equation y 3y + 2y = 0.

9. Determine whether or not the given set is a subspace of M2,2:

(a) The set of all 2 2 matrices, the sum of whose entries is zero.

(b) The set of all 2 2 matrices whose determinant is zero.

10. Determine whether or not the given set is a subspace of Mn,n, the space of all square matrices of size n.

(a) The diagonal matrices of order n.

(b) The matrices of order n with trace equal to 0. (The trace of a square matrix is the sum of thediagonal elements.)

11. (Intersections of subspaces) Let H and K be subspaces of a vector space V . Prove that the intersectionK H is a subspace of V .

Linear combinations and spanning sets, AR5.2

12. Let u = (1, 0,1) and v = (2, 1, 1).

(a) Write w1 = (1, 2,1) as a linear combination of u and v.

(b) Show that w2 = (1, 1, 1) cannot be written as a linear combination of u and v.

(c) For what value of c is the vector (1, 1, c) a linear combination of u and v?

13. Determine whether the given set spans the given vector space.

(a) In R2:{(

12

),

(34

)}.

(b) In R3:

20

1

, 31

2

, 11

1

, 73

5

.

(c) In Z32:

10

1

, 11

0

, 01

1

, 11

1

.14. Determine which of the following sets span R3.

(a) {(1, 2, 3), (1, 0, 1), (0, 1, 2)}

(b) {(1, 1, 2), (3, 3, 1), (1, 2, 2)}

25


15. Find spanning sets for the following subspaces of R3:

(a) {(2a, b, 0) : a, b R}

(b) {(a+ c, c b, 3c) : a, b, c R}

(c) {(4a+ d, a+ 2b, c b) : a, b, c, d R}

16. Find a set that spans the subspaces given in question 6.

17. (Hard) (Function spaces) Which of the following lie in the complex vector space spanned by f = eix

and g = eix?

(a) cosx (b) sinx (c) cosx+ 3i sinx(d) sin 2x (e) coshx

Linear independence and bases, AR5.3, AR5.4

18. In this question let S = {v1,v2,v3,v4,v5} be vectors in R3 and let A be the 3 5 matrix with the ithcolumn given by the vector vi. Suppose that the row-reduced echelon form of A is1 2 0 1 00 0 1 3 0

0 0 0 0 1

Are the following sets linearly dependent or independent? If linearly dependent, express one vector as alinear combination of the others.

(a) {v1,v2,v3} (b) {v1,v3,v4} (c) {v1,v4,v5} (d) {v3,v4,v5}

19. Determine whether or not the following sets of vectors are linearly independent:

(a) {(2,3, 1,5), (0, 1, 2, 2), (1,2, 3, 0)}

(b) {(1, 0, 2,3), (0,4, 1, 1), (2, 2, 0,1), (1,2,1, 3)}

20. Determine whether the following sets are linearly dependent or linearly independent.

(a) {(1, 2), (0, 2), (1, 0), (1, 1)}

(b) {(1, 2), (3,1)}

(c) {(1, 0, 1), (1, 1, 0), (0, 1, 1))}

(d) {(2, 0, 0, 0), (2, 1, 0, 0), (1, 3,2, 0), (1,2, 4,3)}

21. Which of the following sets of vectors in C3 are linearly independent?

(a) {(1 i, 1, 0), (2, 1 + i, 0), (1 + i, i, 0)}

(b) {(1, 0,i), (1 + i, 1, 1 2i), (0, i, 2)}

(c) {(i, 0, 2 i), (0, 1, i), (i,1 4i, 3)}

22. Which of the following sets of vectors in Z32 are linearly independent?

(a) {(1, 1, 0), (1, 1, 1), (0, 0, 1)}

(b) {(1, 0, 0), (1, 1, 0), (1, 1, 1)}

(c) {(1, 1, 0), (1, 0, 1), (0, 1, 1)}

23. (Hard) Show that the vectors (1, a, a2), (1, b, b2), (1, c, c2) are linearly independent if a, b, c are distinct (i.e.,

26


a 6= b, a 6= c and b 6= c). Can you generalise this?

24. In each part determine whether or not the given set forms a basis for the indicated (sub)space.

(a) {(1, 2, 3), (1, 0, 1), (0, 1, 2)} for R3

(b) {(1, 1, 2), (3, 3, 1), (1, 2, 2)} for R3

(c) {(1,1, 0), (0, 1,1)} for the subspace of R3 consisting of all (x, y, z) such that x+ y + z = 0.

(d) {(1, 1, 0), (1, 1, 1)} for the subspace of R3 consisting of all (x, y, z) such that y = x+ z.

(e) {(1, 1, 0), (0, 1, 1)} for the subspace of Z32 consisting of all (x, y, z) such that x+ y + z = 0.

25. Which of the following sets of vectors are bases for R3?

(a) {(1, 0, 0), (2, 2, 0), (3, 3, 3)}

(b) {(2,3, 1), (4, 1, 1), (0,7, 1)}

26. Find a basis for and the dimension of the subspace of Rn spanned by the following sets.

(a) {(0, 1,2), (3, 0, 1), (3, 2,3)} (n = 3)

(b) {(1, 3), (1, 2), (7, 6)} (n = 2)

(c) {(1, 2, 0, 4), (3, 1,1, 2), (5, 3, 1, 6), (7, 0,2, 0)} (n = 4)

27. For each of the following sets choose a subset which is a basis for the subspace spanned by the set. Thenexpress each vector that is not in the basis as a linear combination of the basis vectors.

(a) (1, 2, 0,1), (2,1, 2, 3), (1,11, 6, 13), (4, 3, 2, 1)

(b) (0,1,3, 3), (1,1,3, 2), (3, 1, 3, 0), (0,1,2, 1)

(c) (1, 2,1), (0, 3, 4), (2, 1,6), (0, 0, 2)

(d) (1, 1, 1, 1), (1, 0, 1, 0), (0, 1, 1, 0), (0, 0, 1, 1) in Z42

[Suggestion: Write down the given vectors as columns.]

28. In each part explain why the given statement is true by inspection.

(a) The set {(1, 0, 3), (1, 1, 0), (1, 2, 4), (0,1,2)} is linearly dependent.

(b) The set {(1,1, 2), (0, 1, 1)} does not span R3.

(c) If the set {v1,v2,v3,v4} of vectors in R4 is linearly independent, then it spans R4.

(d) The set {(0, 1,1, 0), (0,1, 2, 0)} is linearly independent, and so it spans the subspace of R4 of allvectors of the form (0, a, b, 0).

29. (a) Show that any four polynomials in P2 are linearly dependent.

(b) Show that two polynomials cannot span P2.

(c) (Hard) Prove that if V and W are three-dimensional subspaces of R5, then V and W must have anonzero vector in common. (Hint: Start with bases for the two subspaces, making six vectors in all.)

30. (Symmetric matrices) A matrix A is symmetric if it is equal to its transpose AT , obtained by inter-

27


changing the rows and columns of A. Let Sn be the set of all symmetric n n matrices. Show that Sn isa subspace of Mn,n and that dimSn = n(n+ 1)/2.

31. (Sums and intersections) Let H and K be subspaces of a vector space V and define

H +K = {h+ k : h H and k K}.

(a) Show that H +K is a subspace of V .

(b) Assume that H and K are each finite dimensional, and that H K = {0}. Show that dim(H+K) =dim(H) + dim(K).

32. (Hard) (Wronskian) Let f and g be differentiable functions from R to R.

(a) Show that f and g are linearly independent if the determinantf(x) g(x)f (x) g(x)

is non-zero for some x in R.

(b) Show that the functions sinx and cosx are linearly independent.

(c) Try to generalise the result of (a) to more than two functions.

Coordinate vectors, AR 5.4

33. Let V be a vector space and suppose that B is a basis for V . Show that any vector in V can be writtenuniquely as a linear combination of elements from B.

34. Find the coordinate vector of v with respect to the given basis B for the vector space V .

(a) v = 2 x+ 3x2,B = {1, x, x2, x3}, V = P3.

(b) v =

[1 2 11 1 2

],B = {Eij |i = 1, 2; j = 1, 2, 3}, V = M2,3.

(c) v = 2 5x,B = {x+ 1, x 1}, V = P1

(d) v =

[2 00 3

],B =

{[1 00 0

],

[0 00 1

]}, V is the vector space of all diagonal 2 2 matrices.

35. (a) Show that the set B = {(2, 2, 2), (3,2, 3), (2,1, 1)} is a basis for R3.

(b) Find the vectors x,y R3 whose coordinates with respect to B are

[x]B =

211

and [y]B = 101

.(c) For each of the following vectors find its coordinates with respect to B:

a = (2,1, 1), b = (1, 0, 5), c = (3,1, 6).

36. Use coordinate vectors to decide whether or not the given set is linearly independent. If it is linearlydependent, express one of the vectors as a linear combination of the others.

(a) {x2 + x 1, x2 2x+ 3, x2 + 4x 3} P2

28


(b)

{[1 21 0

],

[0 11 1

],

[1 01 2

]}in M2,2

Row, column and solution spaces, AR5.5

37. In each part find a basis for and the dimension of the indicated subspace.

(a) The solution space of the homogeneous linear system:

x1 2x2 + x3 = 0x2 x3 + x4 = 0

x1 x2 + x4 = 0

(b) The solution space of

x1 3x2 + x3 x5 = 0x1 2x2 + x3 x4 = 0x1 x2 + x3 2x4 + x5 = 0

(c) The subspace of R4 of all vectors of the form (x,y, x 2y, 3y).

(d) The solution space in Z52 of

x1 + x2 + x3 + x5 = 0x1 + x3 + x4 = 0x1 + x2 + x3 + x4 + x5 = 0

38. Explain why the dimensions of the row space and the column space of a matrix are the same. What arethey for the following matrices?

(a)

1 2 12 1 1

(b) 1 0 11 0 1

(c)

1 1 3

0 1 1

1 1 0

2 1 1

39. Find a basis for the column space for each matrix of question 38.

40. Find a basis for the row space for each matrix of question 38.

41. Find a basis for the solution space for each matrix of question 38.

42. Find bases for the following subspaces of R3.

(a) The set of vectors lying in the plane 2x y z = 0.

(b) The set of vectors on the line x/2 = y/3 = z/4.

43. Is

[6 81 8

]a linear combination of the matrices

A =

[4 02 2

], B =

[1 12 3

], and C =

[0 21 4

]?

44. Express 97x15x2 as a linear combination of p1 = 2+x+4x2, p2 = 1x+3x2, and p3 = 3+2x+5x2.

45. Determine whether or not the given set is linearly independent. If the set is linearly dependent, write oneof its vectors as a linear combination of the others.

29


(a) {1, 1 + x, 1 + x+ x2} in P2

(b) {1 + x2, 1 + x+ 2x2, x+ x2} in P2

(c)

{[1 00 1

],

[0 10 1

],

[0 01 1

],

[0 00 1

]}in M2,2

(d) {1, sin2 x, cos2 x} in C(,), the vector space of all continuous functions from R = (,) toR.

(e)

2 0 00 1 0

0 0 1

,2 0 00 1 0

0 0 1

in M3,3(f)

{[1 00 1

],

[0 10 1

],

[0 01 1

],

[1 01 0

]}in M2,2 over Z2

46. Determine whether or not each set in question 45 generates (that is, spans) the indicated vector space.

47. Determine whether or not each set in question 45 is a basis for the indicated vector space.

48. Find the dimension of the given vector space:

(a) The subspace of M2,2 consisting of all diagonal 2 2 matrices.

(b) The subspace of M2,2 consisting of all 2 2 matrices whose diagonal entries are zero.

(c) The subspace of P3 consisting of all polynomials p(x) = a0 + a1x+ a2x2 + a3x3 with a2 = 0.

49. Determine whether the given set of vectors spans the given vector space.

(a) In P2: {1 x, 3 x2}.

(b) In M2,2:

{(2 10 0

),

(0 02 1

),

(3 10 0

),

(0 03 1

)}

(c) In M2,2 over Z2:{[

1 00 1

],

[0 11 0

],

[0 10 1

],

[0 01 1

],

[1 01 0

],

[1 10 0

]}50. Which of the following sets of vectors are bases for P2?

(a) {1 3x+ 2x2, 1 + x+ 4x2, 1 7x} (b) {1 + x+ x2, x+ x2, x2}

51. Show that the following set of vectors is a basis for M2,2:

{[3 63 6

],

[0 11 0

],

[0 812 4

],

[1 01 2

]}

30


Problem Sheet 6: Inner Product Spaces

Inner Product Spaces AR 6.1

1. In R2, for x =(x1x2

)and y =

(y1y2

), define x,y = x1y1 + 3x2y2.

Show that x,y is an inner product on R2.

2. In R2, let x,y = x1y1 x2y2. Is this an inner product? If not, why not?

3. Verify that the operationx,y = x1y1 x1y2 x2y1 + 3x2y2

where x = (x1, x2) and y = (y1, y2) is an inner product in R2.

4. Decide which of the suggested operations on x = (x1, x2, x3) and y = (y1, y2, y3) in R3 define an innerproduct:

(a) x,y = x1y1 + 2x2y2 + x3y3, (b) x,y = x21y21 + x22y22 + x23y23 ,

(c) x,y = x1y1 x2y2 + x3y3, (d) x,y = x1y1 + x2y2.

5. Decide which of the operations p, q on real polynomials p(x) = a0+a1x+a2x2 and q(x) = b0+b1x+b2x2define an inner product:

(a) p, q = a0b0 + a1b1 + a2b2 (b) p, q = a0b0(c) p, q =

10p(x)q(x) dx

6. (a) If U =

[u1 u2u3 u4

]and V =

[v1 v2v3 v4

]are any two 2 2 matrices, then

U, V = u1v1 + u2v2 + u3v3 + u4v4

defines an inner product on M2,2.

Compute U, V if U =[

3 24 8

]and V =

[1 31 1

].

(b) If p = a0 + a1x+ a2x2 and q = b0 + b1x+ b2x

2 are any two vectors in P2, then

p, q = a0b0 + a1b1 + a2b2

is an inner product on P2. Compute p, q if p = 2 + x+ 3x2 and q = 4 7x2.

7. For the vectors x = (1, 1, 0), y = (0, 1, 0) in R3 compute the norms x and y as well as the anglebetween x and y using the following inner products.

(a) x,y = x1y1 + x2y2 + x3y3 (b) x,y = x1y1 + 3x2y2 + x3y3

8. Let P2 have the inner product defined in question 6b. If p = 2 + 3x+ 2x2, find p.

9. Let M2,2 have the inner product defined in question 6a. If A =

[2 53 6

], find A.

10. Let P2 have the inner product defined in question 6b. If p = 3 x+ x2, q = 2 + 5x2, find d(p, q).

11. Consider M2,2 with the inner product defined in question 6a. If A =

[2 69 4

]and B =

[4 71 6

], find

d(A,B).

31


12. (Parallelogram law) Prove that the following holds for all vectors x,y in every inner product space.

x + y2 + x y2 = 2x2 + 2y2

Give a geometric interpretation of this result.[Hint: consider lengths of edges connecting vertices of a parallelogram with sides given by the vectors xand y.]

13. (Quadratic forms) Let A be a real invertible n n matrix. Show that

x,y yTATAx = (Ay)T (Ax)

defines an inner product in Rn, where x and y are column vectors in Rn. What happens when A is notinvertible? (Note: MT is the transpose of a matrix M , obtained by interchanging the rows and columnsof M .)

14. (Hard) By choosing appropriate vectors in the Cauchy-Schwartz inequality, prove that

(a1 + . . .+ an)2 6 n(a21 + . . .+ a

2n)

for all real numbers a1, . . . , an. When does equality hold?

Orthogonality and Orthonormal Bases, AR6.2, AR6.3

15. In each part determine whether the given vectors are orthogonal with respect to the Euclidean innerproduct (i.e., the usual dot product).

(a) u = (1, 3, 2), v = (4, 2,1) (b) u = (0, 3,2, 1), v = (5, 2,1, 0)

16. Consider R2 and R3 each with the Euclidean inner product. In each part find the cosine of the anglebetween u and v.

(a) u = (1,3), v = (2, 4) (b) u = (1, 5, 2), v = (2, 4,9)

17. Show that p = 1 x+ 2x2 and q = 2x+ x2 are orthogonal with respect to the inner product in question6b.

18. Let A =

[2 11 3

]. Which of the following matrices are orthogonal to A with respect to the inner product

in question 6a?

(a)

1 10 1

(b)2 1

5 2

19. Show that in every inner product space: v + w is orthogonal to v w if and only if v = w. Give a

geometric interpretation of this result.

20. Let x,y be an inner product on a vector space V , and let e1, e2, . . . , en be an orthonormal basis for V .Prove:

(a) For each x V , x = x, e1e1 + x, e2e2 + + x, enen;

(b) 1e1 + 2e2 + + nen, 1e1 + 2e2 + + nen = 11 + 22 + + nn;

(c) x, y = x, e1y, e1+ + x, eny, en.

32


21. Express the given vector as a linear combination of the vectors in the orthonormal basis (with respect tothe dot product). {

(1

3,2

3,

2

3), (2

3,

1

3,

2

3), (

2

3,

2

3,

1

3)

}.

(a) x = (1, 2, 3) (b) y = (1, 0, 1)

22. (Hard) (Pythagorass theorem) Let u1, u2, . . . , un be orthogonal vectors in an inner product space Vand let x =

x, x be the norm induced by the inner product on V . Show that

u1 + u2 + + un2 = u12 + u22 + + un2.

23. Use the Gram-Schmidt procedure to construct orthonormal bases for the subspaces of Rn spanned by thefollowing sets of vectors (using the dot product):

(a) (1, 0, 1, 0), (2, 1, 1, 1), (1,1, 1,1)

(b) (2, 2,1, 0), (2, 3, 1,2), (3, 4, 5,2)

(c) (1,2, 1, 3,1), (0, 6,2,6, 0), (4,2, 2, 6,4)

24. Find the orthogonal projection of (x, y, z) onto the subspace of R3 spanned by the vectors

(a) (1, 2, 2), (2, 2, 1); (b) (1, 2, 1), (0, 1, 2).

25. (Hard) Let V be a finite dimensional real vector space with inner product , , and let W be a subspaceof V . Then the orthogonal complement of W is defined as

W = {v V : v, w = 0 for all w W}.

Prove the following:

(a) W is a subspace of V .

(b) W W = {0}.

(c) dimW + dimW = dimV .

26. Find the least square polynomial of the specified degree n for the given data points.

(a) {(0, 0), (1, 0), (2, 1), (3, 3), (4, 5)}, n = 1

(b) {(2, 2), (1, 1), (0,1), (1, 0), (2, 3)}, n = 2

27. A maths lecturer was placed on a rack by his students and stretched to lengths L = 1.7, 2.0 and 2.3 metreswhen forces of F = 1, 2 and 4 tonnes were applied. Assuming Hookes law L = a + bF , find his normallength a by least squares.

28. A firm that manufactures widgets finds the daily consumer demand d(x) for widgets as a function of theirprice x is as in the following table:

x 1 1.5 2 2.5 3d(x) 200 180 150 100 25

Using least squares polynomials, approximate the daily consumer demand

(a) By a linear function.

(b) By a quadratic function.

33


29. Let P2 be the vector space of polynomials of degree at most two with the inner product

p, q = 11p(x)q(x) dx.

Obtain an orthonormal basis for P2 from the basis {1, x, x2} using the Gram-Schmidt process.

30. (Hard) (Fourier series) Show that the infinite set{ 12,

1

sinnx,1

cosnx : n = 1, 2, . . .}

is an orthonormal set in the vector space C[0, 2] of real continuous functions on the interval [0, 2]equipped with the inner product

f, g = 20

f(x)g(x) dx.

31. (Hard) Find the least square approximation of f(x) = 1 + x in the vector space C[0, 2] equipped withthe inner product of the preceding problem by

(a) a trigonometric polynomial of order 2 or less.

(b) a trigonometric polynomial of order k or less.

[Suggestion: Use the orthonormal basis of the preceding problem.]

Note that

xn sinx dx = xn cosx+n

xn1 cosx dx and

xn cosx dx = xn sinxn

xn1 sinx dx.

34


Problem Sheet 7: Linear Transformations

Linear Transformations AR8.13

1. Show that each of the following functions is a linear transformation:

(a) S : R2 R2, S(x, y) = (2x y, x+ y),

(b) T : R3 M2,2 given by T (x, y, z) =[y zx 0

].

2. Determine whether or not the given function is a linear transformation, and justify your answer.

(a) F : R3 R2, F (x, y, z) = (0, 2x+ y) (b) K : R2 R4, K(x, y) = (x, sin y, 2x+ y)

3. Let v1, v2, and v3 be vectors in a vector space V and T : V R3 a linear transformation for whichT (v1) = (1,1, 2), T (v2) = (0, 3, 2), and T (v3) = (3, 1, 2). Find T (2v1 3v2 + 4v3).

4. For the linear transformations of R2 into R2 given by the following matrices: (i) Sketch the image ofthe rectangle with vertices (0, 0), (2, 0), (0, 1), (2, 1). (ii) Describe the geometric effect of the lineartransformation.

(a)

0 11 0

(b)0 0

1 0

(c)1 1

0 0

(d)

1 0a 1

(e)b 0

0 c

(f) 153 4

4 3

5. (Geometric transformations) Find the matrix of the following linear transformations of R2.

(a) rotation by 34 (b) rotation by 2

(c) reflection in the line y = x (d) reflection in the xaxis

6. In each part, find a single matrix that performs the indicated succession of operations.

(a) Compresses by a factor of 12 in the x-direction, then expands by a factor of 5 in the y-direction.

(b) Reflects about y = x, then rotates about the origin through an angle of .

(c) Reflects about the y-axis, then expands by a factor of 5 in the x-direction, and then reflects abouty = x.

7. Determine whether or not v1 = (2, 0, 0, 2) or v2 = (2, 2, 2, 0) is in the kernel of the linear transformationT : R4 R3 given by T (x) = Ax where

A =

1 2 1 11 0 1 12 4 6 2

.

8. Determine whether or not w1 = (1, 3, 1) or w2 = (1,1,2) is in the image of the linear transformationgiven in question 7.

9. For the linear transformation given in question 7, find the nullity of T and give a basis for the kernel ofT . Is the transformation injective?

10. For the linear transformation given in question 7, find the rank of T and give a basis for the image of T .

35


Is the transformation surjective?

11. For the linear transformation T find (i) its standard matrix, (ii) a basis for the kernel and (iii) a basis forthe image.

(a) T

xy

=x+ y

3y

(b) Tx1

x2

x3

=

x1 + x2 x32x1 + x2

(c) T

xy

=x+ 2y

y

x y

(d) Tx1

x2

x3

=

3x1 x2 6x32x1 + x2 + 5x33x1 + 3x2 + 6x3

12. Determine whether or not the given linear transformation is invertible. If it is invertible, compute its

inverse.

(a) T : R3 R3 given by T (x, y, z) = (x+ z, x y + z, y + 2z)

(b) T : R2 R2 given by T (x, y, z) = (3x+ 2y,6x 4y)

(c) T : R2 R2 an anticlockwise rotation around the origin through an angle of .

(d) T : R2 R2 a reflection in the line through the origin which forms an angle with the x-axis.

13. Show that the transformation

T

xyz

=x+ yy + zz + x

is invertible and find its inverse.

14. Consider the matrix

A() =

cos sin 0sin cos 00 0 1

.(a) Evaluate detA().

(b) Interpret geometrically the effect of multiplying a vector by A().

(c) Show that A()A() = A( + ) and interpret this result.

(d) Use the previous part to find the inverse of A(). How does this compare this with the transposeA()T ?

15. Find the matrices of the transformations T which project a point (x, y, z) onto the following subspaces ofR3. Show by two methods that each transformation is idempotent (i.e., T T = T ).

(a) The z-axis.

(b) The straight line x = y = 2z.

(c) The plane x+ y + z = 0.

36


16. (Computer graphics) One of the most important applications of linear transformations is computergraphics where we wish to view 3-dimensional objects (for example a crystal) on a 2-dimensional screen.The screen is the xy-plane. The aim is to rotate the crystal and orthogonally project it onto the xy-planeto obtain different views of it. We consider 3 possible rotations:

A rotation of round the x-axis using the matrix Rx =

1 0 00 cos sin 0 sin cos

A rotation of round the y-axis using the matrix Ry =

cos 0 sin 0 1 0 sin 0 cos

A rotation of round the z-axis using the matrix Rz =

cos sin 0sin cos 00 0 1

The matrix used to orthogonally project the object onto the computer screen is P =

[1 0 00 1 0

].

The crystal has vertices A : (0, 0, 1), B : (1, 0, 0), C : (0, 1, 0), D : (1, 0, 0), E : (0,1, 0), F : (0, 0,1)with edges the line segments AB, AC, AD, AE, FB, FC, FD, FE, BC, CD, DE and EB.

y

x

z

A

B

C

D

E

F

Projecting onto the xy-plane, we find P (A) = A is given by Px =

[1 0 00 1 0

]001

= [00

],

so that A = (0, 0). In the same way B = (1, 0), C = (0, 1), D = (1, 0), E = (0,1) and F = (0, 0).Connecting up the line segments appropriately we find the view on the computer screen is

y

xB

C

D

E

A& F

Draw the picture that would appear on the computer screen if:

(a) the crystal is rotated 45 around the x-axis before projection,

(b) the crystal is rotated 45 around the x-axis and then 30 around the z-axis,

(c) the crystal is rotated 45 around the x-axis, 30 around the z-axis and then 60 around the y-axis.

You might like to think about whether or not orthogonal projection is the best way to project from 3-Dto 2-D.

37


Matrix Representations, AR8.4

17. The following functions are all linear transformations. Use them in questions 17 to 19.

K(x, y, z) = (x, x+ y, x+ y + z) L(x, y, z) = (2x y, x+ 2y)S(x, y, z) = (z, y, x) T (x, y) = (2x+ y, x+ y, x y, x 2y)

Find the matrix that represents each of the following linear transformations (with respect to the standardbases).

(a) K (b) L (c) S (d) T

18. (Combining linear transformations) Find the indicated linear transformation if it is defined. If it isnot defined, explain why it is not.

(a) LK(= L K) (b) TL(= T L) (c) S2 (d) K + S (e) T 2

19. Find the matrix which represents those linear transformations in question 17 which exist. (Use yourresults from the previous question.)

20. Let Pn denote the vector space of all real polynomials in the variable x of degree 6 n. Let T : P2 P3denote the function defined by multiplication by x: T (p(x)) = xp(x). In other words, T (a + bx + cx2) =ax+ bx2 + cx3.

(a) Show that T is a linear transformation.

(b) Find the matrix of T with respect to the standard bases {1, x, x2} for P2 and {1, x, x2, x3} for P3.

21. Let T : P2 P2 be the linear transformation defined by T (p(x)) = p(2x+ 1), that is,

T (a0 + a1x+ a2x2) = a0 + a1(2x+ 1) + a2(2x+ 1)

2.

Find [T ]B with respect to the basis B = {1, x, x2}.

22. Find the matrix A that represents the linear transformation T with respect to the bases B and B.

(a) T : R3 M2,2 given by

T (x, y, z) =

[y zx 0

]where B = {e1, e2, e3} and B = {Eij |i = 1, 2; j = 1, 2} (i.e. the standard basis for M2,2).

(b) T : P3 P3 given by

T (a0 + a1x+ a2x2 + a3x

3) = (a0 + a2) (a1 + 2a3)x2

where B,B = {1, x, x2, x3}.

23. Compute ker(T ) for each linear transformation in 22. Which ones are one-to-one?

24. Compute Im(T ) for each linear transformation in 22. Which ones are onto?

25. Let V be the vector space of all real 2 2 matrices. Let T : V R2 be the map defined by

T

([a11 a12a21 a22

])=[2 1

] [a11 a12a21 a22

]=[2a11 a21 2a12 a22

].

(a) Show that T is a linear transformation.

(b) Find bases for the kernel and image of T . Deduce the rank and nullity of T .

38


(c) Find the matrix of T with respect to the basis{[1 00 0

],

[0 10 0

],

[0 01 0

],

[0 00 1

]}of V and the standard basis of R2.

26. Let S : P2 P3 be defined as follows. For each p(x) = a2x2+a1x+a0, define S(p) = 13a2x3+ 12a1x

2+a0x.Find the matrix A that represents S with respect to the bases B = {1, x, x2} and B = {1, x, x2, x3} (Thelinear transformation S gives the integral of p(x), with the constant term equal to zero.)

27. Use the matrix of question 26 to find the integral of p(x) = 1 x+ 2x2.

Change of basis, AR8.5,6.5

28. (a) Find the transition matrix P from B to C, where

B = {(1,2, 1), (0, 3, 2), (1, 0,1)} and C = {(1, 0, 0), (0, 1, 0), (0, 0, 1)}.

(b) Use P to find [x]B (the coordinate vector of x with respect to B) if

(a) x = (3,2, 5) (b) x = (2, 7, 4)

29. Verify that the given set B is a basis for Rn. Compute the change of basis matrix for each of the bases,and use it to find the coordinate vector of v with respect to B.

(a) B = {(1, 2), (1,2)},v = (1, 3), (n = 2)

(b) B = {(1, 1, 1), (1, 0, 1), (1, 1, 0)},v = (3,1, 1), (n = 3)

30. Let T : Rn Rn be given by T (x) = Ax where A is the given matrix.

(a) A =

[1 10 1

]Find the matrix that represents T with respect to the basis B of question 29a.

(b) A =

1 1 32 1 11 2 0

Find the matrix that represents T with respect to the basis B of question 29b.

31. Let T : R3 R3 be given by T (x) = T (x, y, z) = (2xy, x+y+z, yz). Find the matrix that representsT with respect to the basis B of question 29b.

32. Write down the matrix of T with respect to B, and compute the matrix of T with respect to B, whereT : R2 R2 is defined by T (x1, x2) = (x1 2x2,x2), B = {u1,u2}, B = {v1,v2}, and u1 = (1, 0),u2 = (0, 1), v1 = (2, 1), v2 = (3, 4).

33. A linear transformation T : R3 R3 has matrix 2 3 01 1 22 0 1

with respect to the standard basis for R3. Find the matrix of T with respect to the basis

B = {(1, 2, 1), (0, 1,1), (2, 3, 2)}.

39


Problem Sheet 8: Eigenvalues and Eigenvectors

Eigenvectors, Eigenvalues, AR7.13, AR9.67

1. Find the eigenvalues and linearly independent eigenvectors of the following matrices.

(a)

2 01 2

(b) 7 2

15 4

(c)1 1

1 3

(d)0 1

1 0

2. Find 1-dimensional subspaces of R2 invariant under the operation of the following matrices:

(a)

1 02 0

(b) 1 23 6

3. (Rotations) Show that there is no line in the real plane R2 through the origin which is invariant under

the transformation whose matrix is

A() =

[cos sin sin cos

],

when is not an integral multiple of . Give a geometric interpretation of this problem commenting onthe case when = k for some k Z.

4. Find the eigenvalues of the given matrix by inspection.

(a)

1 2 0

0 3 1

0 0 4

(b)

1 0 0 0

2 3 0 0

4 5 6 0

7 8 9 10

5. Prove that for an invertible matrix A, is an eigenvalue of A if and only if 1 is an eigenvalue of A

1.What relationship holds between the eigenvectors of A and A1?

6. Find the eigenvalues and linearly independent eigenvectors for the following matrices.

(a)

2 3 6

0 5 6

0 1 0

(b)

2 1 0

0 2 0

0 0 3

(c)

5 8 12

6 10 12

6 10 13

(d)

2 2 2

1 1 2

1 2 3

7. For each matrix find all eigenvalues and a basis for each eigenspace.

(a)

3 1 1

2 4 2

1 1 3

(b)

1 1 0

0 1 0

0 0 1

40


Cayley-Hamilton Theorem

8. Verify the Cayley-Hamilton Theorem for the following matrices.

(a)

3 61 2

(b)

0 1 0

0 0 1

1 3 3

9. Use the Cayley-Hamilton Theorem to calculate the inverse of the matrix

0 1 00 0 11 3 3

.10. For each matrix, find a non-zero polynomial satisfied by the matrix.

(a)

2 51 3

(b)

1 4 3

0 3 1

0 2 1

Diagonalization

11. Decide which of the matrices A in questions 1 and 6 above are diagonalizable and, if possible, find aninvertible matrix P and a diagonal matrix D such that P1AP = D.

12. Let D be a diagonal matrix, D = diag (1, . . . , s) (i.e., Dii = i, Dij = 0 for i 6= j). Prove by inductionthat, for each positive integer n,

Dn = diag (n1 , . . . , ns ).

13. Let A be a matrix such that A = PDP1, where D is diagonal. Prove by induction that, for each positiveinteger n,

An = PDnP1.

14. Use the results of the preceding two problems to find A5, where A is

(a)

3 22 2

(b)

9 18 24

7 20 24

7 21 25

(c) 18

8 1 27 5

0 18 14 6

0 2 18 6

0 8 8 8

[Hint: The matrix has eigenvalues 3, 2,1 in (b), and 1, 2,1,2 in (c).]

15. Suppose the nth pass through a manufacturing process is modelled by the linear equations xn = Anx0,

where x0 is the initial state of the system and

A =1

5

[3 22 3

].

Show that

An =

[12

12

12

12

]+ (

1

5)n

[12

12

1212

].

Then, with the initial state x0 =

[p

1 p

], calculate limn xn.

41


16. (Hard) (Fibonacci numbers) The Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, . . . is described by the difference

equation Fk+2 = Fk+1 + Fk and the initial conditions F0 = 0, F1 = 1. Writing uk =

[Fk+1Fk

], show that

uk+1 = Auk where

A =

[1 11 0

].

Solve for uk in terms of u0 =

[10

]and show that

Fk =15

(1 +52

)k

(1

5

2

)k .Hence find the limit as k of the ratio Fk+1/Fk.

17. Two companies, Lemon and LIME, introduce a new type of computer. At the start, their shares of themarket are 60% and 40%. After a year, Lemon kept 85% of its customers and gained 25% of LIMEscustomers; LIME gained 15% of Lemons customers and kept 75% of its customers. Assume that the totalmarket is constant and that the same fractions shift among the firms every year.

(a) Write down the market share shift as a system of linear equations.

(b) Express the shift in matrix form and find the transition matrix A.

(c) Find the market shares after 5 and 10 years.

(d) Show that the market eventually reaches a steady state, and give the limit market shares.

18. (Hard) Prove by induction that the following statements are true for all positive integers n.

(a) If is an eigenvalue of the square matrix A, then n is an eigenvalue of the matrix An.

(b) (A1)n = (An)1 if A is invertible.

(c) (AT )n = (An)T for any square matrix A.

(d) (AB)n = AnBn, provided that AB = BA.(What happens when the matrices A and B do not commute?)

19. Determine whether or not the given matrix A is orthogonal.

(a)

13

23

23

23

13

23

23

23

13

(b)

0 1 0

1 0 1

0 1 0

20. Show that the rotation matrix A =

[cos sin sin cos

]is orthogonal.

42


21. (Symmetric matrices) For each symmetric matrix A below find a decomposition A = PDPT , where Pis orthogonal and D diagonal.

(a)

7 2 0

2 6 2

0 2 5

(b)2 0 36

0 3 0

36 0 23

(c)

1 1 0

1 1 0

0 0 0

(d)

4 2 2

2 4 2

2 2 4

22. Let A be an orthogonal matrix. Show that detA = 1.

23. Prove that if A,B are orthogonal n n matrices, then so are A1 and AB.

24. (Conics) Name and sketch the following conics:

(a) x2 + y2 = 4 (b) x2 + 4y2 = 1

(c) 3x2 4y2 = 4 (d) x 4y2 = 0

(Quadrics) Name and sketch the following quadrics:

(a) 2x2 + y2 + 6z2 3 = 0 (b) x2 + 5y2 z2 = 4

(c) 6x2 2y2 z2 2 = 0 (d) x2 + y2 + z2 25 = 0

25. For each matrix A in question 21 classify and sketch the quadric xTAx = 1.

26. Classify and sketch the following quadrics. For each give directions of the principal axes and the lengthsof semiaxis:

(a) 4x2 + 4y2 8z2 10xy 4xz 4yz = 1

(b) 8x21 7x22 + 8x23 + 8x1x2 2x1x3 + 8x2x3 = 9

27. The given equations contain no linear terms. Reduce them to standard form and identify the quadricsurfaces that they represent.

(a) x2 + y2 + z2 2xy 2xz 2yz = 2

(b) 3x2 + 3y2 + 3z2 3xy 4yz = 66

(c) x2 + y2 + z2 + 6xy + 8yz = 0

(d) x2 + 2

2xy = 4

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Problem Sheet 9: Functions of Two Variables

Level Curves, Tangent Planes and Derivatives

1. Identify the level sets I - VI and the graph surfaces A - F corresponding to the equations (a) - (f).

(a) z = sinx2 + y2 (b) z = sin (x) (c) z =

1

x2 + 4y2

(d) z = 1 2x 2y (e) z = sin (x) sin (y) (f) z = x2 y2

Graphs: Level Sets:

z

x

A

y

z

yx

By

x

Iy

x

II

z

yx

C z

yx

D

y

x

III y

x

IV

z

yx

E z

yx

Fy

x

V y

x

VI

2. Draw some level curves for the following functions:

(a) f(x, y) = xy (b) f(x, y) = x2 y2

(c) f(x, y) = xy (d) f(x, y) =x+ y

(e) f(x, y) = y cosx (f) f(x, y) = x y2

44


3. Functions of two variables are able to be visualized as surfaces in R3. For example, to visualize f(x, y),create the surface z = f(x, y). This is not available in higher dimensional problems.

(a) Find and describe the level curves for each of the following functions defined on R2.

(i) f(x, y) = x, (ii) f(, ) = (iii) f(, ) = 2 + 42.

(b) Describe and sketch the surface z = f(x, y):

(i) f(x, y) =

1 x2 y2 (ii) f(x, y) = 16y2 x2

(iii) f(x, y) =

72 + 4x2 9y2 (iv) f(x, y) = 6 2x 3y

4. (a) In each case, what is the largest subset of R2 on which the following formulae define functions, andwhat is the corresponding range?

(i) f(x, y) =x2 + 16y4

exp (x); (ii) f(u, v) = log(u2 v2)

(b) Where a is a positive constant, what is the largest subset of R3 on which f(, , ) = log(a2 2 2 2) defines a function, and what is the corresponding range?

5. (i) Find the first and second partial derivatives of the following functions:

(a) f(x, y) = 3x2 + 2xy + y5 (b) f(x, y) = sin(x) sin(y)

(c) f(x, y) = (x3 y2)2 (d) f(r, s) =r2 + s2

(e) f(x, y) = yex (f) f(u,w) = arctan(u/w)

(g) f(r, s, t) = r2e2s cos (t)

(ii) Verify that2w

xy=

2w

yxif w = x2 cos

(x

y

).

(iii) If f(x, y) = logx2 + y2, show that f is harmonic, that is

2f

x2+2f

y2= 0.

(iv) Show that v = sin (akt) sin (kx) satisfies the wave equation2v

t2= a2

2v

x2.

6. Peculiar Property. If f(x, y) = xey/x, show that

xfx + yfy = f

and thatx2fxx + 2xyfxy + y

2fyy = 0.

45


7. Let f(x, y) =x2y

x4 + y2.

(a) Let m be any constant, and consider the line y = mx. Find the limit of f(x, y) as we approach (0, 0)along this line. That is, find lim

t0f(t,mt).

(b) Now consider the line x = 0. Find the limit of f(x, y) as we approach (0, 0) along this line.

(c) Now consider the curve y = x2. Find the limit of f(x, y) as we approach (0, 0) along this curve.

(d) What do you conclude about whether f(x, y) has a limit at (0, 0)?

8. Find the following limits or show that they dont exist.

(a) lim(x,y)(1,1)

x yx+ y

(b) lim(x,y)(0,0)

3x+ 2y

x+ y

(c) lim(x,y)(1,3)

(x+ y)2 + 2xy

5 cos2 (x) sin2 (xy)

(d) lim(x,y)(3,2)

(x2 9)(xy + 2)(y2 4y + 4)(x 3)(y 2)2(x+ 1)

(e) lim(x,y)(0,0)

sin(x2 + y2)

x2 + y2

9. By using the chain rule finddf

dtfor each of the following:

(a) f(x, y) = x2 + y2 where x = 2 sin(t) and y = 3 cos(t),

(b) f(x, y) = x log(y) where x = t2 + 1 and y = et,

(c) f(x, y) = exy where x = 2et and y = et.

10. Suppose f(x, y) = x3 + 2xy and that x = t, y = 1 + t2.

(a) Express f as a function of t only and then differentiate with respect to t to obtaindf

dt.

(b) Calculate the partial derivatives fx and fy and hence evaluatedf

dt.

11. (Hard)

(a) Let w = x2 + y2 + z2, x = et cos t, y = et sin t, z = et.

Find dw/dt by (a) substitution and (b) the chain rule.

(b) Suppose that w = log(x2 + y2 + 2z), x = r + s, y = r s, z = 2rs.

Find w/r and w/s by the chain rule.

46


12. A moving particle has position (sin t, cos t, t) at time t.

(i) Find the velocity and acceleration of the particle at time t.

(ii) Show that the particle moves on the cylinder x2 + y2 = 1.

(iii) Sketch the curve traced out by the particle as t varies.

(iv) (Hard) If the temperature at position (x, y, z) is T (x, y, z) = ex2y2z2 , find the rate of change of

the particles temperature at time t.

13. Find equations of the tangent plane and the normal line to the graph of the given function at the indicatedpoint:

(a) z = 4x2 + y2 78 at (2, 1,61) (b) z = x sin(y) at(1, 12, 1

)(c) z = 4xy at (0, 0, 0) (d) z = 4x2 y2 at (5,8, 36)

(e) z = 2ex cos (y) at(0, 13, 1

)14. For the following functions f , find the gradient vector f at the indicated point.

(a) f(x, y) = x3 + 2xy + xy2 at (0, 1) (b) f(x, y) = sin(x) cos(y) at(, 14

)(c) f(x, y) = 3x2 xy + y2 at (2, 1) (d) (Hard) f(x, y, z) = 2z3 3(x2 + y2)z at

(1, 1, 1)

(e) (Hard) f(x, y, z) = e3x+4y cos(5z) at(0, 0, 16

)15. Find the directional derivatives of the following functions at the indicated point in the direction specified.

(a) f(x, y) = arcsin(x/y) at (1, 2) in the direction /4 anticlockwise from the x-axis

(b) f(x, y) =x2 + y2 at (3, 4) in the direction away from the origin towards the point (3, 4)

(c) f(x, y) = x2 3xy + y2 at (1, 1) in the direction from (1, 1) towards (1, 2)

(d) f(x, y) = x3y2 at (1, 2) in the direction 13 clockwise from the positive x-axis

(e) g(x, y) = sin (xy) at(16 , )

in the direction of the unit vector(35 ,

45

)(f) h(x, y) = x2e2y at (4, 3) towards the point (6, 0)

(g) (Hard) F (x, y, z) = xy + yz + xz at (1,1, 2) in the direction of the vector (10, 11,2)

16. Steepest ascent/descent.

(a) Find the direction in which the function f(x, y) = x3 +y26xy increases and decreases most rapidlyat the point (3, 3).

(b) Find the direction in the xy-plane one should travel, starting from the point (1, 1), to obtain themost rapid rate of decrease of f(x, y) = (x+ y 2)2 + (3x y 6)2.

(c) In which direction in the xy-plane is the directional derivative of the function f(x, y) = (x2y2)/(x2+y2) at the point (1, 1) equal to zero?

(d) The temperature at any point (x, y) of a heated plate is 100(x2 + 2y2 + 1)1. At the point (1, 2),in what direction(s) is the rate of change of temperature (i) zero, and (ii) greatest? How are thesedirections related?

47


17. Check that the level curve x3 + 2xy + x2y2 + y3 = 17 passes through the point (1, 2).

(a) If f(x, y) = x3 + 2xy + x2y2 + y3, calculate f at the point (1, 2) and hence write down the slope ofthe normal to the curve at (1, 2).

(b) Using implicit differentiation calculate the slope of the above curve at (1, 2).

(c) Check that the product of your answers to parts (a) and (b) is equal to 1.

18. Prove that a normal vector to the surface f(x, y) =xy at any point on the surface is perpendicular to

the line joining the point to the origin.

19. (Hard) Find the points on the hyperboloid x2 2y2 4z2 = 16 at which the tangent plane is parallel tothe plane 4x 2y + 4z = 5.

20. If the length of the hypotenuse of a right-angled triangle is 4.98 cm, and the length of one of the othersides is 3.02 cm, find, without using a calculator, a reasonable approximation to the length of the thirdside.

21. The pressure P of an ideal gas of volume V and absolute temperature T are related by

PV = T,

where is a constant that depends on the gas. If the temperature increases by 5%, what approximatepercentage change in volume is necessary to keep the pressure constant?

Critical Points

22. Find the Hessian matrices for the functions:

(a) f(x, y) = x3 + 3xy y3 (b) f(x, y) = x4 + y3 + 32x 9y

(c) f(x, y) = cos (x) + cos (y) (d) (Hard) f(x, y, z) = x2 + y2 3xy z

23. Stationary Points.

(a) Show that f(x, y) = 4x2 + xy y2 has a maximum at (0, 0).

(b) Show that g(x, y) =ex+y

x3y2has a minimum at the point (3, 2).

24. Find the maxima, minima and saddle points of the following functions:

(a) x3 + 3xy y3 (b) x4 + y3 + 32x 9y (c) cos (x) + cos (y)

25. Least Squares. A satellite television repeater station is to be located at P (x, y) so that the sum of thesquares of the distances from the three towns A,B and C is a minimum. The three towns are located atthe positions A(0, 0), B(2, 6) and C(10, 0). Find the coordinates of the repeater station.

26. A company produces two types of surfboard, x thousand of type A and y thousand of type B per year. Ifthe revenue R and cost C equations for the year are (in millions of dollars),

R(x, y) = 2x+ 3y,

C(x, y) = x2 2xy + 2y2 + 6x 9y + 5

determine how many of each type of surfboard should be made per year to maximize the profit. What isthe maximum profit?

48


27. (Hard)

(a) Find the shortest distance from the point (2, 1,1) to the plane 4x 3y + z = 5.

(b) Find the points on the graph of xy3z2 = 16 that are closest to the origin.

(c) Find the point on the sphere x2 + y2 + z2 = 9 that is closest to the point (2, 3, 4).

Double Integrals

28. Integrating Partial Derivatives. Find the partial integrals of the function

f(x, y) = 3x2 + 2xy + y5

(i) with respect to x, and (ii) with respect to y.

29. Integrating Partial Derivatives In each case, find the most general function f(x, y) that satisfies the givenfirst-order partial differential equations.

(a)f

x= 4x(x2 y2) + cosx f

y= 4y(x2 y2)

(b)f

x= 2y sec2(2xy)

f

y= 2x sec2(2xy)

(c)f

x= 2e2x cos y 3ex sin 4y + 1 f

y= e2x sin y 12ex cos 4y

30. Do

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