Undergraduate MathematicalEconomicsLecture 1
Yu Ren
WISE, Xiamen University
September 15, 2014
math
Courses Description and RequirementLinear Algebra
Outline
1 Courses Description and Requirement
2 Linear Algebra
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Course Outline
mathematical techniques used ineconomics coursesMathematics for Economists by Simon andBlume
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Course Outline
mathematical techniques used ineconomics coursesMathematics for Economists by Simon andBlume
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Course Outline
15 weeks, including one midterm examgrading policy
quiz: 30 %midterm exam:30%final exam:40%
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Course Outline
15 weeks, including one midterm examgrading policy
quiz: 30 %midterm exam:30%final exam:40%
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Quiz
7 quizzes in totaleach has two questions, covering thematerials of the previous lecturerequired to finish it within 20 minutes
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Quiz
7 quizzes in totaleach has two questions, covering thematerials of the previous lecturerequired to finish it within 20 minutes
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Quiz
7 quizzes in totaleach has two questions, covering thematerials of the previous lecturerequired to finish it within 20 minutes
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Office hours
my office hours: 2:30-4:00pm TuesdayTA office hours: TBAany courses-related questions can beasked
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Some Questions
1 Why economics need mathematics?2 What is the focus of this course?3 How to achieve good performances in this
course
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Some Questions
1 Why economics need mathematics?2 What is the focus of this course?3 How to achieve good performances in this
course
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Some Questions
1 Why economics need mathematics?2 What is the focus of this course?3 How to achieve good performances in this
course
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Linear Algebra
Linear AlgebraChapter 6, 7, 8, 9
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Linear Equation
an equation is linear if
a1x1 + a2x2 + · · ·+ anxn = b
ai : parameters, xi : variables (i = 1,2, · · · ,n)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Linear Equation
For examplex1 + 2x2 = 32x1 − 3x2 = 8
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Linear Systems
a11x1 + a12x2 + · · · + a1nxn = b1a21x1 + a22x2 + · · · + a2nxn = b2
... +... + · · · +
... =...
am1x1 + am2x2 + · · · + amnxn = bm
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Example: Tax Benefits
Before tax profits: 100,00010% of after-tax profits to Red Crossstate tax of 5% of its profits after Red Crossdonationfederal tax of 40% of its profits afterdonation and state tax
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Example: Tax Benefits
Question: How much does the companypay in state taxes, federal taxes and RedCross?
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Example: Answer
C: (Red Cross); S: (State taxes); F: (Federaltaxes)
C = (100,000− S − F ) ∗ 0.1 (1)S = (100,000− C) ∗ 0.05 (2)
F = (100,000− C − S) ∗ 0.4 (3)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
General Questions
Questions:1 Does a solution exist?2 How many solutions are there?3 Is there an efficient algorithm that computes
actual solutions?
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
General Questions
method to find the answers1 substitutions2 elimination of variables3 matrix methods
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Substitution
Taught in high school classSteps:
1 m equations, n variables2 solve xn in terms of the other variables3 substitute this expression for xn into the other
equations4 a new system of m − 1 equations and n − 1
variables5 repeat the process
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Substitution
Taught in high school classSteps:
1 m equations, n variables2 solve xn in terms of the other variables3 substitute this expression for xn into the other
equations4 a new system of m − 1 equations and n − 1
variables5 repeat the process
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Substitution
Taught in high school classSteps:
1 m equations, n variables2 solve xn in terms of the other variables3 substitute this expression for xn into the other
equations4 a new system of m − 1 equations and n − 1
variables5 repeat the process
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Substitution
Taught in high school classSteps:
1 m equations, n variables2 solve xn in terms of the other variables3 substitute this expression for xn into the other
equations4 a new system of m − 1 equations and n − 1
variables5 repeat the process
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Substitution
Taught in high school classSteps:
1 m equations, n variables2 solve xn in terms of the other variables3 substitute this expression for xn into the other
equations4 a new system of m − 1 equations and n − 1
variables5 repeat the process
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Elimination
Taught in high school classSteps:
1 m equations, n variables2 multiplying both sides of an equation by a
nonzero number3 adding one equation to another equation in
order to eliminate one variable
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Elimination
Taught in high school classSteps:
1 m equations, n variables2 multiplying both sides of an equation by a
nonzero number3 adding one equation to another equation in
order to eliminate one variable
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Elimination
Taught in high school classSteps:
1 m equations, n variables2 multiplying both sides of an equation by a
nonzero number3 adding one equation to another equation in
order to eliminate one variable
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Elimination
Example (Page 126):
x1 − 0.4x2 − 0.3x3 = 130−0.2x1 + 0.88x2 − 0.14x3 = 74−0.5x1 − 0.2x2 + 0.95x3 = 95
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Matrix Methods: Elementary RowOperations
augmented matrix: add on a columncorresponding to the right-hand side insystem
A =
a11 a12 · · · a1n b1a21 a22 · · · a2n b2... ... · · · ... ...
am1 am2 · · · amn bm
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Matrix Methods: Elementary RowOperations
interchange two rows of a matrixchange a row by adding to it a multiple ofanother rowmultiply each element in a row by the samenonzero number
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Definitions
leading zeros: a row of a matrix is said tohave k leading zeros if the first k elementsof the row are all zeros and the k + 1element of the row is not zero
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Definitions
row echelon form: a matrix is in rowechelon form if each row has more leadingzeros than the row preceding it. Example: 1 2 3
0 2 30 0 3
row echelon form can be obtained byelementary row operations
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Rank
Rank: the rank of a matrix is the number ofnonzero rows in its row echelon form.a matrix→ row echelon form→ rank
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Rank
Let A be the coefficient matrix of somelinear systems and let A be thecorresponding augmented matrix. Then thissystems has a solution if and only ifrank(A)=rank(A).
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Matrix Algebra: Addition
a11 · · · a1n... aij
...ak1 · · · akn
+
b11 · · · b1n... bij
...bk1 · · · bkn
=
a11 + b11 · · · a1n + b1n... aij + bij
...ak1 + bk1 · · · akn + bkn
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Matrix Algebra: Subtraction
a11 · · · a1n... aij
...ak1 · · · akn
− b11 · · · b1n
... bij...
bk1 · · · bkn
=
a11 − b11 · · · a1n − b1n... aij − bij
...ak1 − bk1 · · · akn − bkn
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Matrix Algebra: scalar multiplication
r
a11 · · · a1n... aij
...ak1 · · · akn
=
ra11 · · · ra1n... raij
...rak1 · · · rakn
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Matrix Algebra: matrix multiplication
We can define the matrix product AB if and onlyif number of column of A = number of rowsof B .(i , j) entry of AB is
(ai1 ai2 · · · aim
)b1jb2j...
bmj
=m∑
h=1
aihbhj
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Laws of Matrix Algebra
Associative Laws:(A+B) +C = A+ (B +C); (AB)C = A(BC)
Commutative Law for addition:A + B = B + ADistributive Laws: A(B + C) = AB + AC;(A + B)C = AC + BC
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Transpose
transpose:a11 a12 · · · a1na21 a22 · · · a2n... ... aij
...ak1 ak2 · · · akn
T
=
a11 a21 · · · ak1a12 a22 · · · ak2... ... aji
...a1n a2n · · · akn
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Transpose
(A + B)T = AT + BT ; (A− B)T = AT − BT ;(AT )T = A; (rA)T = rAT ; (AB)T = BT AT
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Special Matrices
Page 160Square Matrix; Column Matrix; Row Matrix;Diagonal Matrix; Upper-Triangular Matrix;Lower-Triangular Matrix; Symmetric Matrix;Permutation MatrixIdempotent Matrix; Nonsingular Matrix
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Inverse
Let A be a n × n matrix. Matrix B is aninverse of A if AB = BA = I. If the matrix Bexists, we say A is invertible.
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Inverse
Theorem 8.5: An n × n matrix can have atmost one inverse.(A−1)−1 = A; (AT )−1 = (A−1)T ; AB isinvertible and (AB)−1 = B−1A−1
How to find the inverse matrix? Hold thisquestion for a while.
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Inverse
Theorem 8.5: An n × n matrix can have atmost one inverse.(A−1)−1 = A; (AT )−1 = (A−1)T ; AB isinvertible and (AB)−1 = B−1A−1
How to find the inverse matrix? Hold thisquestion for a while.
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Inverse
Theorem 8.5: An n × n matrix can have atmost one inverse.(A−1)−1 = A; (AT )−1 = (A−1)T ; AB isinvertible and (AB)−1 = B−1A−1
How to find the inverse matrix? Hold thisquestion for a while.
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Determinants
det(a) = a
det(
a11 a12a21 a22
)= a11a22 − a12a21
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Determinants
det(a) = a
det(
a11 a12a21 a22
)= a11a22 − a12a21
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Minor and Determinant
Let A be an n × n matrix. Let Aij be the(n − 1)× (n − 1) submatrix obtained bydeleting row i and column j from A. Thenumber Mij = det(Aij) is called the (i , j)minor of A and the scalar Cij = (−1)i+jMij iscalled the (i , j) cofactor of A.
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Minor and Determinant
Determinant of an n× n matrix A is given by
det(A) = a11C11 + a12C12 + · · ·+ a1nC1n
Example 9.2 (Page 192)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Minor and Determinant
Determinant of an n× n matrix A is given by
det(A) = a11C11 + a12C12 + · · ·+ a1nC1n
Example 9.2 (Page 192)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Uses of the Determinant
For any n × n matrix A, let Cij denotes the(i , j)th cofactor of A. The n × n matrixwhose (j , i) entry is Cij is called the adjointof A, denoted by adj(A).
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Uses of the Determinant
A−1 = 1det(A)adj(A)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Uses of the Determinant
(Cramer’s Rule) the unique solutionx = (x1, · · · , xn) of the n × n system Ax = bis xi =
det(Bi)det(A) , where Bi is the matrix A with
the right-hand side b replacing the i thcolumn of AExample 9.3 (Page 195)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Uses of the Determinant
(Cramer’s Rule) the unique solutionx = (x1, · · · , xn) of the n × n system Ax = bis xi =
det(Bi)det(A) , where Bi is the matrix A with
the right-hand side b replacing the i thcolumn of AExample 9.3 (Page 195)
Yu Ren Undergraduate Mathematical Economics Lecture 1
math
Courses Description and RequirementLinear Algebra
Some Properties of Determinant
det(AT ) = det(A)det(AB) = (det(A))(det(B))
det(A + B) 6= det(A) + det(B)
Yu Ren Undergraduate Mathematical Economics Lecture 1