Linear Algebra

216 Chapter 3. Maps Between Spaces
Distribute and regroup on the v’s.
=(gi,1h1,1 + gi,2h2,1 + ···+ gi,rhr,1) · v1
Finish by recognizing that the coefficient of each vj
+ ···+(gi,1h1,n + gi,2h2,n + ···+ gi,rhr,n) · vn
gi,1h1,j + gi,2h2,j + ···+ gi,rhr,j
matches the definition of the i, j entry of the product GH. QED
The theorem is an example of a result that supports a definition. We can
picture what the definition and theorem together say with this arrow diagram
(‘w.r.t.’ abbreviates ‘with respect to’).
↗h H
Vw.r.t. B
Ww.r.t. C
g◦h
−→
GH
↘ g
G
Xw.r.t. D
Above the arrows, the maps show that the two ways of going from V to X,
straight over via the composition or else by way of W , have the same effect
�v g◦h
↦−→ g(h(�v)) �v h
↦−→ h(�v)
g
↦−→ g(h(�v))
(this is just the definition of composition). Below the arrows, the matrices
indicate that the product does the same thing—multiplying GH into the column
vector Rep B(�v) has the same effect as multiplying the column first by H and
then multiplying the result by G.
Rep B,D(g ◦ h) =GH =Rep C,D(g)Rep B,C(h)
The definition of the matrix-matrix product operation does not restrict us
to view it as a representation of a linear map composition. We can get insight
into this operation by studying it as a mechanical procedure. The striking thing
is the way that rows and columns combine.
One aspect of that combination is that the sizes of the matrices involved is
significant. Briefly, m×r times r×n equals m×n.
2.7 Example This product is not defined
�
−1 2
��
0 0
�
0
0 10 1.1 0 2
because the number of columns on the left does not equal the number of rows
on the right.