LINEAR MODELS AND MATRIX ALGEBRA Part 2 Chapter

LINEAR MODELS AND MATRIX ALGEBRA - Part 2 Chapter 4 Alpha Chiang, Fundamental Methods of Mathematical Economics 3 rd edition

Vector Operations l 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 u’v will be of dimension 1 x 1. l Example 1 - 2 x 1, 1 x 3, 2 x 3.

Vector Operations l Example 2. 1 x 2, 2 x 1, 1 x 1 l As written, u’v 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], [3][7]=[21] ¡ a scalar product

Vector Operations l Example 3. - Given a row vector u’ = [3 6 9], find u’u. Since u is merely a column vector, with elements of u’ arranged vertically, we have, l Note that the product u’u gives the sum of squares of the elements of u (a scalar).

Linear Dependence l A set of vectors v 1, …, 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. l are linear dependent because v 3 is a linear combination of v 1 and v 2:

Linear Dependence l Example 5. v 1’ =[5 12] and v 2’ = [10 24] are linearly dependent because 2 v 1’= 2[5 12] = [10 24] = v 2’ or 2 v 1’-v 2’ = 0 l A set of m-vectors v 1, …, vn is linearly dependent if and only if there exists a set of scalars k 1, …, kn (not all zero) such that

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

Commutative, Associative, And Distributive Laws l Matrix Addition: commutative and associative l Commutative law : A+B=B+A

Commutative, Associative, And Distributive Laws l Associative law: (A+B) + C = A + (B+C)

Commutative, Associative, And Distributive Laws l Matrix Multiplication: not commutative l Example:

Commutative, Associative, And Distributive Laws l Example: Let u’ be a 1 x 3 (a row vector); then the corresponding column vector u must be 3 x 1. The product u’u will be 1 x 1 but the product uu’ will be 3 x 3. Thus obviously, u’u ≠ uu’. l Exceptions: ¡A is a square matrix and B is an identity matrix ¡A is the inverse of B, A = B-1 ¡scalar multiplication: k. A=Ak

Commutative, Associative, And Distributive Laws l Associative Law: (AB)C=A(BC)=ABC Conformability condition: A is mxn, B is nxp, C is pxq l Distributive Law: A(B+C) = AB + AC [pre-multiplication by A] (B+C)A = BA + CA [post-multiplication by A]