Linear Algebra, 2020a

264 Chapter Three. Maps Between Spaces
We finish this subsection by recognizing the change of basis matrices as a
familiar set.
1.5 Lemma A matrix changes bases if and only if it is nonsingular.
Proof For the ‘only if’ direction, if left-multiplication by a matrix changes
bases then the matrix represents an invertible function, simply because we can
invert the function by changing the bases back. Because it represents a function
that is invertible, the matrix itself is invertible, and so is nonsingular.
For ‘if’ we will show that any nonsingular matrix M performs a change of
basis operation from any given starting basis B (having n vectors, where the
matrix is n×n) to some ending basis.
If the matrix is the identity I then the statement is obvious. Otherwise
because the matrix is nonsingular Corollary IV.3.23 says there are elementary
reduction matrices such that R r ···R 1 · M = I with r 1. Elementary matrices
are invertible and their inverses are also elementary so multiplying both sides of
that equation from the left by R −1 r , then by R −1 r−1 , etc., gives M as a product
−1 of elementary matrices M = R 1 ···R −1 r .
We will be done if we show that elementary matrices change a given basis to
another basis, since then R −1 −1
r changes B to some other basis B r and R r−1
changes B r to some B r−1 , etc. We will cover the three types of elementary
matrices separately; recall the notation for the three.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
c 1... c 1...
c 1... c 1...
⎛ ⎞ ⎛ ⎞
c 1... c 1...
c i
c j
c i
c i
M i (k)
c i
=
kc
⎜ . i
P
⎝
⎟ ⎜
i,j . .
=
. .
C i,j (k)
. .
=
. .
. ⎠ ⎝
⎟
. ⎠
c j
c i
c j
kc i + c j
⎜ .
c n c n ⎝
⎟ ⎜ .
. ⎠ ⎝
⎟ ⎜ .
. ⎠ ⎝
⎟ ⎜
. ⎠ ⎝
⎟
. ⎠
c n c n c n
Applying a row-multiplication matrix M i (k) changes a representation with
respect to 〈⃗β 1 ,...,⃗β i ,...,⃗β n 〉 to one with respect to 〈⃗β 1 ,...,(1/k)⃗β i ,...,⃗β n 〉.
c n
⃗v = c 1 · ⃗β 1 + ···+ c i · ⃗β i + ···+ c n · ⃗β n
↦→ c 1 · ⃗β 1 + ···+ kc i · (1/k)⃗β i + ···+ c n · ⃗β n = ⃗v
The second one is a basis because the first is a basis and because of the k ≠ 0
restriction in the definition of a row-multiplication matrix. Similarly, leftmultiplication
by a row-swap matrix P i,j changes a representation with respect