1. LINEAR MODELS AND MATRIX ALGEBRA - Part 2 Chapter 4 Alpha Chiang, Fundamental Methods of Mathematical Economics 3 rd 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 u’v will be of dimension 1 x 1. Example 1- 2x1, 1x3, 2x3.

3. Vector Operations Example 2. 1x2, 2x1, 1x1 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

4. Vector Operations 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, Note that the product u’u gives the sum of squares of the elements of u (a scalar).

5. Linear Dependence A set of vectors v 1, …,v n 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. v 1 ’ =[5 12] and v 2 ’ = [10 24] are linearly dependent because 2v 1 ’= 2[5 12] = [10 24] = v 2 ’ or 2v 1 ’-v 2 ’ = 0 A set of m-vectors v 1, …,v n is linearly dependent if and only if there exists a set of scalars k 1, …, k n (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 additiona + b = b + a Commutative law of multiplicationab = ba Associative law of addition(a+b) + c = a+ (b+c) Associative law of multiplicationab (c) = a(bc) Distributive law a (b+c) = ab + ac

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

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

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 u’u will be 1x1 but the product uu’ will be 3x3. Thus obviously, u’u ≠ uu’. Exceptions:  A is a square matrix and B is an identity matrix  A is the inverse of B, A = B -1  scalar multiplication: kA=Ak

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