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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 onetoone....
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 Fall '14

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