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4.8 Change of Basis and Similarity 251
4.8 CHANGE OF BASIS AND SIMILARITY

By their nature, coordinate matrix representations are basis dependent. However,
it’s desirable to study linear transformations without reference to particular bases
because some bases may force a coordinate matrix representation to exhibit
special properties that are not present in the coordinate matrix relative to other
bases. To divorce the study from the choice of bases it’s necessary to somehow
identify properties of coordinate matrices that are invariant among all bases—
these are properties intrinsic to the transformation itself, and they are the ones
on which to focus. The purpose of this section is to learn how to sort out these
basis-independent properties.
The discussion is limited to a single finite-dimensional space V and to linear
operators on V. Begin by examining how the coordinates of v ∈V change as
the basis for V changes. Consider two different bases

B = {x 1 , x 2 ,..., x n } and B = {y 1 , y 2 ,..., y n } .

It’s convenient to regard B as an old basis for V and B as a new basis for V.

Throughout this section T will denote the linear operator such that
T(y i )= x i for i =1, 2,...,n. (4.8.1)

T is called the change of basis operator because it maps the new basis vectors
in B to the old basis vectors in B. Notice that [T] B =[T] B =[I] BB . To see

this, observe that
n n n

x i = α j y j =⇒ T(x i )= α j T(y j )= α j x j ,
j=1 j=1 j=1

which means [x i ] B =[T(x i )] , so, according to (4.7.4),
B

[T] B = [T(x 1 )] B [T(x 2 )] B ··· [T(x n )] B = [x 1 ] B [x 2 ] B ··· [x n ] B =[T] B .

The fact that [I] BB =[T] B follows because [I(x i )] B =[x i ] B . The matrix

(4.8.2)
P =[I] BB =[T] B =[T] B
will hereafter be referred to as a change of basis matrix. Caution! [I] BB is
not necessarily the identity matrix—see Exercise 4.7.16—and [I] BB =[I] B B .
We are now in a position to see how the coordinates of a vector change as
the basis for the underlying space changes.