Linear system, REF, RREF, nullspace

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University of Toronto Scarborough
Sophie Chrysostomou

Linear Systems and Matrices Linear Systems DEFINITION: An m n linear system of equations is a system of m linear equations in n variables: a11 +1a x 12.2. + a x 1n n = b 1 a x + a x + ... + a x = b 21 1 22 2 2n n 2 . . . . . . am1 x1+ a m2x 2 ... + a mn xn = b m where x ,1x arn the unknowns and aij are constant real numbers for all i = 1,2, ,m, and j = 1,2,n. EXAMPLE: 2x1+ 3x 2 = 9 x1 2x 2 = 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. 3. Replace an equation by that equation plus a multiple of another equation. www.notesolution.comDEFINITION: 1. An m n matrix is an ordered rectangular array, of real numbers, with m rows and n columns. a11 a12 a1n a21 a22 a2n . . . . . . . . a a a m1 m2 mn 2. We may denote the matrix by giving it a name, say A, and write A = [a ], where a is the entry in the ihrow and j thcolumn of A. ij ij b1 b2 3. An m 1 matrix . is called a column vector in R . . bm 4. A 1 n matrix c1 c2 cn is called a row vector in R . 5. A matrix is called a square matrix if the number of its rows and the number of its columns are equal. 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 a = 0 for all i j ij 1 for all i = j 8. A square matrix A is called an upper triangular matrix if a = 0 for all i > j ij 9. A square matrix A is called a lower triangular matrix if aij 0 for all i < j 2 2011 by Sophie Chrysostomou
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