HW05-solutions (2)

HW05-solutions (2) - khan(sak2454 HW05 gilbert(57245 This...

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khan (sak2454) – HW05 – gilbert – (57245) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – fnd all choices beFore answering. 001 10.0 points Let T : R 2 R 2 be the transFormation T ( x ) = x 1 b 3 - 1 B + x 2 b - 4 2 B . Determine T ( x ) when x = b 2 1 B . 1. T ( x ) = b 3 - 1 B 2. T ( x ) = b 1 - 1 B 3. T ( x ) = b 3 0 B 4. T ( x ) = b 1 0 B 5. T ( x ) = b 2 - 1 B 6. T ( x ) = b 2 0 B correct Explanation: By matrix-vector multiplication, x 1 u + x 2 v = [ u v ] b x 1 x 2 B . Thus T ( x ) = [ u v ] b x 1 x 2 B = b 3 - 4 - 1 2 Bb x 1 x 2 B , For each x . Consequently, when x = b 2 1 B , T ( x ) = b 3 - 4 - 1 2 2 1 B = b 2 0 B . 002 10.0 points IF T : R 2 R 2 is the linear transFormation such that T pb x 1 x 2 BP = x 1 b - 2 2 B + x 2 b - 1 1 B , determine T ( x ) when x = b 1 3 B . 1. T ( x ) = b - 5 4 B 2. T ( x ) = b - 4 4 B 3. T ( x ) = b - 4 5 B 4. T ( x ) = b - 5 5 B correct 5. T ( x ) = b 1 3 B Explanation: When x = b 1 3 B = b 1 0 B + 3 b 0 1 B , then T ( x ) = T pb 1 0 BP + 3 T pb 0 1 BP = b - 2 2 B + 3 b - 1 1 B . Consequently, T ( x ) = b - 2 2 B + b - 3 3 B = b - 5 5 B . 003 10.0 points IF A is an m × n matrix, then the range oF the transFormation T A : x A x is R m . True or ±alse?

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khan (sak2454) – HW05 – gilbert – (57245) 2 1. FALSE correct 2. TRUE Explanation: When A is m × n , the matrix product A x is only defned ±or x in R n , and the product is then a vector in R m . Thus T A : x A x maps R n to R m . So the Range T A is the set { T A ( x ) : x in R n } o± vectors in R m . But this need not be all R m .
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