Question
Asked Sep 23, 2019
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14) Let T: R
be a linear transformation such that
R
T(x, y,z)= (x + y,x-
y+z, y2z) . Find the trace of T.
a) 5
b) -1
c) 0
d) 7
e) 2
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14) Let T: R be a linear transformation such that R T(x, y,z)= (x + y,x- y+z, y2z) . Find the trace of T. a) 5 b) -1 c) 0 d) 7 e) 2

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Expert Answer

Step 1

Let e1 = (1, 0, 0), e2 = (0, 1, 0) and e3 = (0, 0, 1) be the standard basis in R3.

Find T(e1), T(e2) and T(e3) as follows.

T(1,0,0)(1+0,1-0+0,0+2(0))
-(1.10)
T(0,1,0) (0 1,0-1+0,1+2(0))
(1-1,1)
T(0,0,1)(0+0,0-0+1,0+2(1))
=(0,1,2)
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T(1,0,0)(1+0,1-0+0,0+2(0)) -(1.10) T(0,1,0) (0 1,0-1+0,1+2(0)) (1-1,1) T(0,0,1)(0+0,0-0+1,0+2(1)) =(0,1,2)

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Step 2

Thus, the matrix T(x, y, z) can ...

101
1
T (x, y,)1
-1 1
0
1
2
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101 1 T (x, y,)1 -1 1 0 1 2

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