HW05-solutions (2)

# xn en x1 t e 1 x2 t e 2 xn t

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Unformatted text preview: ty, T (x) = T (x1 e1 + x2 e2 + . . . + xn en ) = x1 T ( e 1 ) + x2 T ( e 2 ) + . . . + xn T ( e n ) . x1 x2 where A is the 2 × 2 matrix standard matrix [T (e1 ) T (e2 )] where T (ej ) is a vector in Rm . So for each x in Rn , T (x) is determined by its eﬀect on e1 , e2 , . . . , en . Consequently, the statement is of T . Now 1 0 1 −2 True or False? Explanation: We can write R2 both as rows and column vectors x1 . (i) (x1 , x2 ) , (ii) x2 e1 = 2 A linear transformation T : Rn → Rm is completely determined by its eﬀect on the columns e1 , e2 , . . . , en of the n × n identity matrix In . 2 01 2. A = 3 A= 1 2 −2 = T (e 2 ) . Consequently, Determine A so that T can be written as the matrix transformation TA : R2 → R2 . 1. A = 2 −2 TRUE . = (1, 0) , e2 = 0 1 = (0, 1) , so that T (1, 0) = (3, 1) = 3 1 008 10.0 points If A is an m × n matrix with m pivot columns, then the linear transformation = T (e 1 ) , TA : Rn → Rm , TA (x) = Ax khan (sak2454) – HW05 – gilbert – (57245) is one-to-one....
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