Page 191 - A Course in Linear Algebra with Applications
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6.2: Linear Transformations and Matrices 175
Hence T is represented with respect to B' by
/ 0 2 0 \
- 1
[/At/ = 0 0 4 .
\ 0 0 0 /
This conclusion is easily checked. An arbitrary element of
2
P 3 (R) can be written in the form / = a(l)+6(2ir) + c(4a; -2).
Then it is claimed that the coordinate vector of T(f) with
respect to the basis B' is
/ 0 2 0 \ / a \ / 2 6 \
0 0 4 6 == 4 c .
\0 0 0/ \ c / W
This is correct since
2
26(1) + 4c(2x) + 0(4a? - 2) = 2b + 8cx
2
= (a{l) + b(2x) + c(4x - 2))'.
Similar matrices
Let A and B be two n x n matrices over a field F; then
B is said to be similar to A over F if there is an invertible
n x n matrix S with entries in F such that
1
B = SAS~ .
Thus the essential content of 6.2.6 is that two matrices which
represent the same linear operator on a finite-dimensional vec-
tor space are similar. Because of this fact it is to be expected
that similar matrices will have many properties in common:
for example, similar matrices have the same determinant. In-
deed if B = SAS~\ then by 3.3.3 and 3.3.5
det(B) = det(S) det(A) det(S)- 1 = det(A).