# Lecture8 - University of Toronto at Scarborough Department...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Winter 2008 Lecture 8 REVIEW of lecture 7 A system of m linear equations in n unknowns is: a11 x1 a21 x1 . . . + a12 x2 + a22 x2 + · · · + a1 n x n + · · · + a2 n x n = b1 = b2 am1 x1 + am2 x2 + · · · + amn xn = bm is equivalent to Ax = b where: A= a11 a21 . . . a12 a22 am1 am2 · · · a1 n · · · a2 n , · · · amn x= x1 x2 . . . , and b = bm xn and A is called the coeﬃcient matrix, The system also corresponds to the augmented matrix: [A|b] = · · · a1 n · · · a2 n b1 b2 . . . am1 am2 · · · amn bm a11 a21 . . . a12 a22 1 b1 b2 . . . . Ax = b corresponds to: a11 a21 a21 a22 + . x1 . x2 . . . . a1 m a2 m ↑ ↑ column 1 of A column 2 of A + ··· an1 an2 . . . xn anm ↑ column n of A = b1 b2 . . . bm therefore Ax = b has a solutions if and only if b is in the span of the columns of A. definition 0.1. (a) Ax = b is consistent if it has at least one solution, otherwise it is called inconsistent. (b) If [H |c] is the result of performing successive elementary row operations to [A|b] then we say that [A|b] and [H |c] are row equivalents and we denote it by [A|b] ∼ [H |c] theorem 0.2. a)If [A|b] ∼ [H |c] then: (i) Ax = b has a solution ⇐⇒ H x = c has a solution. (ii) If the solution set of Ax = b exists, then it is the same as the solution set of H x = c. b) If [A|b] ∼ [H |c] and [H |c] is in REF then: (i) H x = c is inconsistent (has no solution) ⇐⇒ [H |c] has a row of zeros before the partition and a nonzero entry after the partition. (ii) If H x = c is consistent and H has a pivot in every column ⇒ the solution is unique. (iii) If H x = c is consistent and H has columns without pivots ⇒ the solution set is inﬁnite with as many ’free variables ’ (or parameters) as non pivotal columns. A REF of a matrix is not unique, where its RREF is. 2 definition 0.3. An Elementary Matrix. An elementary matrix is the result of performing exactly one elementary row operation to an identity matrix. example 0.4. example 0.5. Through examples, guess the relationship between the matrix B , obtained by performing one elementary row to a matrix A, and E and A, where E is the the elementary matrix that corresponds to that elementary row operation performed on I . 3 011 example 0.6. Reduce the matrix A = 2 2 1 to a RREF H , using 133 only one elementary row operation at a time. Find H in terms of A and the elementary matrices corresponding to the elementary row operations performed. 4 ...
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