Page 66 - Linear Algebra Done Right

Chapter 3. Linear Maps
52
the number of rows of the second matrix.
As an example of matrix multiplication, here we multiply together
You should find an matrices only when the number of columns of the first matrix equals
example to show that a 3-by-2 matrix and a 2-by-4 matrix, obtaining a 3-by-4 matrix:
matrix multiplication is  1 2   10 7 4 1 
not commutative. In  3  6 5 4 3 =  26 19 12 5  .



other words, AB is not 4  2 1 0 −1
5 6 42 31 20 9
necessarily equal to BA,
even when both are Suppose (v 1 ,...,v n ) is a basis of V.If v ∈ V, then there exist unique
defined. scalars b 1 ,...,b n such that
3.12 v = b 1 v 1 +· · ·+ b n v n .
The matrix of v, denoted M(v), is the n-by-1 matrix defined by
 
b 1
 . 
3.13 M(v) =  . .  .
 
b n
Usually the basis is obvious from the context, but when the basis needs

to be displayed explicitly use the notation M v, (v 1 ,...,v n ) instead
of M(v).
n
For example, the matrix of a vector x ∈ F with respect to the stan-
dard basis is obtained by writing the coordinates of x as the entries in
n
an n-by-1 matrix. In other words, if x = (x 1 ,...,x n ) ∈ F , then
 
x 1
 . 
M(x) =  . .  .
 
x n
The next proposition shows how the notions of the matrix of a linear
map, the matrix of a vector, and matrix multiplication fit together. In
this proposition M(Tv) is the matrix of the vector Tv with respect to
the basis (w 1 ,...,w m ) and M(v) is the matrix of the vector v with re-
spect to the basis (v 1 ,...,v n ), whereas M(T) is the matrix of the linear
map T with respect to the bases (v 1 ,...,v n ) and (w 1 ,...,w m ).
3.14 Proposition: Suppose T ∈L(V, W) and (v 1 ,...,v n ) is a basis
of V and (w 1 ,...,w m ) is a basis of W. Then

M(Tv) =M(T)M(v)
for every v ∈ V.