LINEAR MODELS AND MATRIX ALGEBRA Part 2

1
LINEAR MODELS AND MATRIX ALGEBRA- Part 2

• Chapter 4
• Alpha Chiang, Fundamental Methods of Mathematical
  Economics
• 3rd edition

2
Vector Operations

• Multiplication of vectors
• An m x 1 column vector u, and a 1 x n row vector
  v, yield a product uv of dimension m x n. On
  the other hand, a 1 x n row vector u and an n x
  1 column vector v, the product uv will be of
  dimension 1 x 1.
• Example 1- 2x1, 1x3, 2x3.

3
Vector Operations

• Example 2. 1x2, 2x1, 1x1
• As written, uv is a matrix, despite the fact
  that only a single element is present.
• 1 x 1 matrices behave exactly like scalars with
  respect to addition and multiplication 4 8
  12, 3721
• a scalar product

4
Vector Operations

• Example 3. - Given a row vector u 3 6 9,
  find uu. Since u is merely a column vector,
  with elements of u arranged vertically, we have,
• Note that the product uu gives the sum of
  squares of the elements of u (a scalar).

5
Linear Dependence

• A set of vectors v1, ,vn is linearly dependent
  if and only if any one of them can be expressed
  as a linear combination of the remaining vectors
  otherwise, they are linearly independent.
• are linear dependent because v3 is a linear
  combination of v1 and v2

6
Linear Dependence

• Example 5. v1 5 12 and v2 10 24 are
  linearly dependent because
• 2v1 25 12 10 24 v2
• or 2v1-v2 0
• A set of m-vectors v1, ,vn is linearly dependent
  if and only if there exists a set of scalars k1,
  , kn (not all zero) such that

7
Commutative, Associative, And Distributive Laws

• In ordinary scalar algebra, additive and
  multiplicative operations obey the commutative,
  associative, and distributive laws
• Commutative law of addition a b b a
• Commutative law of multiplication ab ba
• Associative law of addition (ab) c a
  (bc)
• Associative law of multiplication ab (c) a(bc)
• Distributive law a (bc) ab ac

8
Commutative, Associative, And Distributive Laws

• Matrix Addition commutative and associative
• Commutative law ABBA

9
Commutative, Associative, And Distributive Laws

• Associative law (AB) C A (BC)

10
Commutative, Associative, And Distributive Laws

• Matrix Multiplication not commutative
• Example

11
Commutative, Associative, And Distributive Laws

• Example Let u be a 1x3 (a row vector) then
  the corresponding column vector u must be 3x1.
  The product uu will be 1x1 but the product uu
  will be 3x3. Thus obviously, uu ? uu.
• Exceptions
• A is a square matrix and B is an identity matrix
• A is the inverse of B, A B-1
• scalar multiplication kAAk

12
Commutative, Associative, And Distributive Laws

• Associative Law (AB)CA(BC)ABC
• Conformability condition
• A is mxn, B is nxp, C is pxq
• Distributive Law
• A(BC) AB AC pre-multiplication by A
• (BC)A BA CA post-multiplication by A