Page 223 - Linear Algebra Done Right

Chapter 10. Trace and Determinant
214
Change of Basis
The matrix of an operator T ∈L(V) depends on a choice of basis
of V. Two different bases of V may give different matrices of T. In this
section we will learn how these matrices are related. This information
will help us find formulas for the trace and determinant of T later in
this chapter.
With respect to any basis of V, the identity operator I ∈L(V) has a
diagonal matrix
 
1 0
 . 
 . .  .
 
0 1
This matrix is called the identity matrix and is denoted I. Note that we
use the symbol I to denote the identity operator (on all vector spaces)
and the identity matrix (of all possible sizes). You should always be
able to tell from the context which particular meaning of I is intended.
For example, consider the equation

M(I) = I;

on the left side I denotes the identity operator and on the right side I
denotes the identity matrix.
If A is a square matrix (with entries in F, as usual) with the same
size as I, then AI = IA = A, as you should verify. A square matrix A
Some mathematicians is called invertible if there is a square matrix B of the same size such
use the terms that AB = BA = I, and we call B an inverse of A. To prove that A has
nonsingular, which at most one inverse, suppose B and B are inverses of A. Then

means the same as

invertible, and B = BI = B(AB ) = (BA)B = IB = B ,
singular, which means
and hence B = B , as desired. Because an inverse is unique, we can use

the same as
the notation A −1 to denote the inverse of A (if A is invertible). In other
noninvertible.
words, if A is invertible, then A −1 is the unique matrix of the same size
such that AA −1 = A −1 A = I.
Recall that when discussing linear maps from one vector space to
another in Chapter 3, we defined the matrix of a linear map with respect
to two bases—one basis for the first vector space and another basis for
the second vector space. When we study operators, which are linear
maps from a vector space to itself, we almost always use the same basis