Linear Algebra

Section IV. Matrix Operations 215
which we recognizing as the result of this matrix-vector product.
⎛
1 · 4+1· 5
= ⎝0 · 4+1· 5
1 · 4+0· 5
1· 6+1· 7
0· 6+1· 7
1· 6+0· 7
1· 8+1· 9
0· 8+1· 9
1· 8+0· 9
⎞
1· 2+1· 3
0· 2+1· 3⎠
1· 2+0· 3
⎛
⎜
⎝
Thus, the matrix representing g◦h has the rows of G combined with the columns
of H.
2.3 Definition The matrix-multiplicative product of the m×r matrix G and
the r×n matrix H is the m×n matrix P , where
pi,j = gi,1h1,j + gi,2h2,j + ···+ gi,rhr,j
that is, the i, j-th entry of the product is the dot product of the i-th row and
the j-th column.
⎛
⎞ ⎛
⎞
.
h1,j
.
⎜ .
⎟ ⎜
GH = ⎜gi,1
gi,2 ⎝
... gi,r⎟
⎜...
h2,j ... ⎟
⎠ ⎜ .
⎝ .
⎟
. ⎠
.
=
⎛
⎞
.
⎜ . ⎟
⎜
⎝
... pi,j ... ⎟
⎠
.
2.4 Example The matrices from Example 2.2 combine in this way.
⎛
1 · 4+1· 5
⎝0
· 4+1· 5
1· 6+1· 7
0· 6+1· 7
1· 8+1· 9
0· 8+1· 9
⎞ ⎛
1· 2+1· 3 9
0· 2+1· 3⎠
= ⎝5
13
7
17
9
⎞
5
3⎠
1 · 4+0· 5 1· 6+0· 7 1· 8+0· 9 1· 2+0· 3 4 6 8 2
2.5 Example
⎛
2
⎝4
8
⎞
0 �
6⎠
1
5
2
⎛
� 2 · 1+0· 5
3
= ⎝4
· 1+6· 5
7
8 · 1+2· 5
⎞ ⎛
2· 3+0· 7 2
4· 3+6· 7⎠
= ⎝34
8· 3+2· 7 18
⎞
6
54⎠
38
2.6 Theorem A composition of linear maps is represented by the matrix product
of the representatives.
Proof. (This argument parallels Example 2.2.) Let h: V → W and g : W → X
be represented by H and G with respect to bases B ⊂ V , C ⊂ W ,andD ⊂ X,
of sizes n, r, andm. For any �v ∈ V ,thek-th component of Rep C( h(�v)) is
hr,j
hk,1v1 + ···+ hk,nvn
and so the i-th component of Rep D( g ◦ h (�v) ) is this.
gi,1 · (h1,1v1 + ···+ h1,nvn)+gi,2 · (h2,1v1 + ···+ h2,nvn)
+ ···+ gi,r · (hr,1v1 + ···+ hr,nvn)
B,D
v1
v2
v3
v4
⎞
⎟
⎠
D