# Lecture4 - University of Toronto at Scarborough Department...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 4 Sophie Chrysostomou 1.3 Linear Systems and Matrices Linear Systems definition 0.1. An m × n linear system of equations is a system of m linear equations in n variables: a11 x1 + a12 x2 + ... + a1n xn a21 x1 + a22 x2 + ... + a2n xn . . . . . . . . . . . . = = . . . b1 b2 . . . am1 x1 + am2 x2 + ... + amn xn = bm where x1 , · · · xn are the unknowns and aij are constant real numbers for all i = 1, 2, · · · , m, and j = 1, 2, · · · n. example 0.2. 2x1 + 3x2 = 9 x1 − 2x2 = 1 In order to solve such a system we perform in some sequence the following three operations: 1. Switch two equations. 2. Multiply an equation by a nonzero constant. 1 3. Replace an equation by that equation plus a multiple of another equation. definition 0.3. 1. An m × n matrix is an ordered rectangular array, of real numbers, with m rows and n columns. a11 a12 ··· a1 n a21 a22 ··· a2 n . . . . . . . . . . . . am1 am2 · · · amn 2. We may denote the matrix by giving it a name, say A, and write A = [aij ], where aij is the entry in the ith row and j th column of A. b1 b2 3. An m × 1 matrix . is called a column vector in Rm . . . bm 4. A 1 × n matrix c1 c2 · · · cn is called a row vector in Rn . 5. A matrix is called a square matrix if the number of its rows and the number of its columns are equal. 2 6. A square matrix A is called a diagonal matrix if aij = 0 for all i = j 7. A square matrix A is called an identity matrix if aij = 0 1 for all for all i=j i=j 8. A square matrix A is called an upper triangular matrix if aij = 0 for all i > j 9. A square matrix A is called a lower triangular matrix if aij = 0 for all i < j 3 ...
View Full Document

## This note was uploaded on 05/22/2010 for the course MATH MATA23 taught by Professor Sophie during the Spring '10 term at University of Toronto.

Ask a homework question - tutors are online