Define the linear transformation T:?3??4 by T(x )=Ax . Find a vector x whose image under T is b
Since A is the standard matrix of the linear transformation T: R3?R4, we have T(e1) = (-2,5,5,2)T, T(e2) = (5,2,-3,1)T and T(e3) = (-5,-2,2,-4)T.
Let M = [A|b] =
-2 |
5 |
-5 |
4 |
5 |
2 |
-2 |
-10 |
5 |
-3 |
2 |
-7 |
2 |
-1 |
-4 |
11 |
The RREF of M is
1 |
0 |
0 |
-2 |
0 |
1 |
0 |
-3 |
0 |
0 |
1 |
-3 |
0 |
0 |
0 |
0 |
It implies that b = -2(-2,5,5,2)T -3(5,2,-3,1)T -3(-5,-2,2,-4)T = -2 T(e1)-3 T(e2)-3 T(e3) = T(-2e1-3e2-3e3) ( as T is linear). Further, -2e1-3e2-3e3 = -2(1,0,0)T-3(0,1,0)T-3(0,0,1)T= (-2,-3,-3)T.
Thus b = T(X), where X =(-2,-3,-3)T. Apparently, X is unique.
Define the linear transformation T:?3??4 by T(x )=Ax . Find a vector x whose image under...
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