Page 193 - A Course in Linear Algebra with Applications

6.2: Linear Transformations and Matrices 177

7. Let B denote the standard basis of R 3 and let B' be the
basis consisting of

Find the matrices that represent the basis changes B —* B'
-
and B' • B.
8. A linear transformation from R 3 to R 2 is defined by

x x - x 2 - x 3
-Xi + X 3

Let B and C be the ordered bases

of R 3 and R 2 respectively. Find the matrix that represents T
with respect to these bases.

9. Explain why the matrices I 1 and ( 1 cannot
be similar.
10. If B is similar to A, prove that A is similar to B.

11. If B is similar to A and C is similar to B, prove that C
is similar to A.
T T
12. If B is similar to A, then B is similar to A ; prove or
disprove.