Linear Algebra

240 Chapter 3. Maps Between Spaces
elementary matrices M = R1 −1 ···Rr −1 . Thus, we will be done if we show
that elementary matrices change a given basis to another basis, for then Rr −1
changes B to some other basis Br, and Rr−1 −1 changes Br to some Br−1,
... , and the net effect is that M changes B to B1. We will prove this about
elementary matrices by covering the three types as separate cases.
Applying a row-multiplication matrix
⎛ ⎞ ⎛ ⎞
c1 c1
⎜ . ⎟ ⎜ . ⎟
⎜ . ⎟ ⎜
⎟ ⎜ . ⎟
Mi(k) ⎜ci
⎟
⎜ ⎟ = ⎜kci⎟
⎜ ⎟
⎜ . ⎟ ⎜
⎝ .
. ⎟
. ⎠ ⎝ . ⎠
cn
changes a representation with respect to 〈 � β1,..., � βi,..., � βn〉 to one with respect
to 〈 � β1,...,(1/k) � βi,..., � βn〉 in this way.
�v = c1 · � β1 + ···+ ci · � βi + ···+ cn · � βn
cn
↦→ c1 · � β1 + ···+ kci · (1/k) � βi + ···+ cn · � βn = �v
Similarly, left-multiplication by a row-swap matrix Pi,j changes a representation
with respect to the basis 〈 � β1,..., � βi,..., � βj,..., � βn〉 into one with respect to the
basis 〈 � β1,..., � βj,..., � βi,..., � βn〉 in this way.
�v = c1 · � β1 + ···+ ci · � βi + ···+ cj � βj + ···+ cn · � βn
↦→ c1 · � β1 + ···+ cj · � βj + ···+ ci · � βi + ···+ cn · � βn = �v
And, a representation with respect to 〈 � β1,..., � βi,..., � βj,..., � βn〉 changes via
left-multiplication by a row-combination matrix Ci,j(k) into a representation
with respect to 〈 � β1,..., � βi − k � βj,..., � βj,..., � βn〉
�v = c1 · � β1 + ···+ ci · � βi + cj � βj + ···+ cn · � βn
↦→ c1 · � β1 + ···+ ci · ( � βi − k � βj)+···+(kci + cj) · � βj + ···+ cn · � βn = �v
(the definition of reduction matrices specifies that i �= k and k �= 0 and so this
last one is a basis). QED
1.5 Corollary A matrix is nonsingular if and only if it represents the identity
map with respect to some pair of bases.
In the next subsection we will see how to translate among representations
of maps, that is, how to change Rep B,D(h) toRepˆ B, ˆ D (h). The above corollary
is a special case of this, where the domain and range are the same space, and
where the map is the identity map.