Practice Questions for Hour Exam I - Gregg

These questions are from an exam I gave at UNL in Summer Session I of 2008

(1) (20 points) Let ~u =

21−1

and ~v =

−112

. Find:

(a) the angle between ~u and ~v

(b) proj~v~u

(c) the vector form of the equation of a plane containing the point P = (2,−4, 5) with directionvectors ~u and ~v.

(d) the equation in parametric form of the line through the point Q = (1, 2, 0) which has directionvector ~v.

(2) (18 points) Mark each of the following statements true or false. If the statement is true, justify youranswer; if it is false, give a counterexample.

(a) Suppose that plane P is parallel to line l. If ~n1 is a normal vector for P and ~n2 is a normalvector for l, then n1 and n2 are parallel.

(b) Every homogeneous system of linear equations is consistent.

(c) If A and B are matrices with AB = 0, then either A = 0 or B = 0.

(3) (5 points) State the Triangle Inequality.

(4) (12 points) Use Gaussian or Gauss-Jordan elimination to solve the system:

a + b− c = 7a + 2b + c = 4

b− c = 3.

(5) (15 points)(a) Give the definition of linearly dependent vectors.

(b) Show that if ~u and ~v are linearly independent, then ~u+~v and ~u−~v are also linearly independent.

(6) (8 points) Balance the equation for the chemical reaction

CO2 + H2O −→ C6H12O6 + O2

(7) (10 points) Let A =

1 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 20 0 0 0 3 4

and B =

0 0 0 0 1 20 0 0 0 −3 10 0 0 0 4 00 0 0 0 1 −11 0 1 0 1 00 1 0 1 0 1

.

Choose a convenient partition for multiplying these matrices (indicate by drawing lines in the matrix).Write A and B in block form, then compute AB.

1

(8) (10 points) Suppose that A−1 =[

1 3−2 1

]and B =

[0 1−1 4

]. Compute (AB)−1.

(9) (10 points) Let A =

1 0 10 2 01 0 2

.

(a) Compute A−1.

(b) Use your result from (a) to solve A~x = ~b, where ~b =

10−1

Bonus: (5 points) Prove the Triangle Inequality.

Additional Practice Questions for Hour Exam I - Gregg

These questions are from an exam I gave here at Augustana in spring semester 2009

(1) (10 points) Let ~u =[

1−1

]and ~v =

[21

].

(a) Draw the coordinate axes relative to the vectors ~u =[

1−1

]and ~v =

[21

].

(b) Write ~w =[

111

]as a linear combination of ~u and ~v.

(2) (12 points) The vector form of a plane P is

xyz

=

42

−1

+ s

110

+ t

011

(a) Does the point (4, 3, 0) lie in the plane? Justify your answer.

(b) Give a vector ~n which is perpendicular to P. Justify your answer.

(3) (5 points) A is a 4× 6 matrix. What are the possible values of rank A? Explain your answer.

(4) (8 points) Below is one step in a student’s work to reduce a matrix to row-echelon form. The stepshown is not an elementary row operation. Show elementary row operations which will achieve thesame result.

[2 23 4

]2R2−3R1−−−−−−→

[2 20 2

]

(5) (8 points) The n× n matrices A,B, and X are invertible. Solve the matrix equation below for X:

(A−1X)−1 = A2B−1

(6) (12 points) Consider the matrices A and R, below. R is the reduced row echelon form of A (youneed NOT verify this!).

A =

1 3 1 0 4 12 6 3 1 10 21 3 0 2 8 31 3 0 0 4 10 0 1 1 2 1

, R = rref(A) =

1 3 0 0 4 00 0 1 0 0 00 0 0 1 2 00 0 0 0 0 10 0 0 0 0 0

(a) Give a basis for col(A).

(b) Give a basis for null(A).

(7) (5 points) A 4×4 elementary matrix E is shown below. What elementary row operation is equivalentto multiplying a 4× n matrix A on the left by E?

1 0 0 00 1 0 30 0 1 00 0 0 1

(8) (10 points) Write, but do not solve, the system of linear equations to determine the currents for thecircuit below.

(9) (10 points) Determine whether each statement is true or false. Explain your answer.

(a) In R3, two lines which are not parallel must intersect in a point.

(b) In R3, two planes which are not parallel must intersect in a line.

(c) The set

101

,

0−1

1

,

000

is linearly independent.

(d) If A is an n× n matrix whose columns sum to ~0, then A is not invertible.

(e) The set S ={[

xy

]: x + y = 2

}a subspace of R2.

(10) (10 points) Use Gaussian elimination (or Gauss-Jordan elimination) to solve the system

3w + 3x + 2y + z = 19−w − y + 2z = −2

w + x + y + z = 7

(11) (10 points) Suppose that the set {~u,~v, ~w} is linearly independent. Prove that the set {~u + ~v,~v + ~w}is linearly independent.

(12) (10 points) Find the inverse of the matrix A =

1 0 12 1 01 1 0

Bonus: Prove or give a counterexample:

If A and B are invertible n× n matrices, then they are row equivalent.