HW05-solutions (2)

# t en with e1 e2 en the columns of the identity

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Unformatted text preview: he m × n matrix A = [T (e1 ) T (e2 ) . . . T (en)] with e1 , e2 , . . . , en the columns of the identity matrix In , then TA (x) = T (x). So T can be identiﬁed with the matrix transformation TA . Consequently, the statement is TRUE . then A x = x1 a 1 + x2 a 2 + · · · + xn a n is a linear combination of the columns of A with weights being the entries in x. Conversely, any linear combination c1 a1 + c2 a2 + · · · + cn an of the columns of A can be written as Ax with c1 c2 x = . . . . cn Thus the range of TA consists of all linear combinations of the columns of A. Consequently, the statement is TRUE . khan (sak2454) – HW05 – gilbert – (57245) 3 and 006 10.0 points Let T : R2 → R2 be the linear transformation such that T (x1 , x2 ) = (3x1 + 2x2 , x1 − 2x2 ) . T (0, 1) = (2, −2) = 3 007 30 1 30 1 01 3. A = 2 3 4. A = 2 1 −2 . 10.0 points 1. FALSE 2. TRUE correct correct In row form Explanation: For each x in Rn , x = In x = [ e 1 e 2 x1 x2 . . . en ] . . . xn = x1 e 1 + x2 e 2 + . . . + xn e n . T (x1 , x2 ) = (3x1 + 2x2 , x1 − 2x2 ) , while in column vector form T x1 x2 =A Thus by lineari...
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## This document was uploaded on 02/26/2014.

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