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LINEAR MODELS AND MATRIX ALGEBRA - Part 2

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

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LINEAR MODELS AND MATRIX ALGEBRA - Part 2

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  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 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 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’= 2[5 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 (a+b) + c = a+ (b+c) Associative law of multiplication ab (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]

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