18.5. THE MATRIX OF A LINEAR TRANSFORMATION 441

Then you do L to this and get

x

 1−11

+ y

 12−1

+ z

 011

=

 x+ y2y− x+ zx− y+ z

Now take the matrix of the transformation times the given column vector. 2 −5 1

−1 4 00 −2 1

 x

yz

=

 2x−5y+ z4y− xz−2y

Is this the coordinate vector of the above relative to the given basis?

(2x−5y+ z)

 101

+(4y− x)

 111

+(z−2y)

 −110

=

 x+ y2y− x+ zx− y+ z

You see it is the same thing.

Now lets find the matrix of L with respect to the usual basis. Let B be this matrix. Thatis, multiplication by B is the same as doing L. Thus

B

 1 1 −10 1 11 1 0

=

 1 1 0−1 2 11 −1 1

Hence

B =

 1 1 0−1 2 11 −1 1

 1 1 −1

0 1 11 1 0

−1

=

 0 0 12 3 −3−3 −2 4

Of course this is a very different matrix than the matrix of the linear transformation withrespect to the funny basis.

What about the situation where different pairs of bases are chosen for V and W? Howare the two matrices with respect to these choices related? Consider the following diagramwhich illustrates the situation.

Fn A2−→

Fm

q2 ↓ ◦ p2 ↓V L

−→W

q1 ↑ ◦ p1 ↑Fn A1

−→Fm

In this diagram qi and pi are coordinate maps as described above. From the diagram,

p−11 p2A2q−1

2 q1 = A1,

where q−12 q1 and p−1

1 p2 are one to one, onto, and linear maps. Thus the effect of thesemaps is identical to multiplication by a suitable matrix.