Linear Algebra, 2020a

234 Chapter Three. Maps Between Spaces
1.4 Theorem Let h, g: V → W be linear maps represented with respect to bases
B, D by the matrices H and G and let r be a scalar. Then with respect to
B, D the map r · h: V → W is represented by rH and the map h + g: V → W is
represented by H + G.
Proof Generalize the examples. This is Exercise 10.
QED
1.5 Remark These two operations on matrices are simple, but we did not define
them in this way because they are simple. We defined them this way because
they represent function addition and function scalar multiplication. That is,
our program is to define matrix operations by referencing function operations.
Simplicity is a bonus.
We will see this again in the next subsection, where we will define the
operation of multiplying matrices. Since we’ve just defined matrix scalar multiplication
and matrix sum to be entry-by-entry operations, a naive thought is
to define matrix multiplication to be the entry-by-entry product. In theory we
could do whatever we please but we will instead be practical and combine the
entries in the way that represents the function operation of composition.
A special case of scalar multiplication is multiplication by zero. For any map
0 · h is the zero homomorphism and for any matrix 0 · H is the matrix with all
entries zero.
1.6 Definition A zero matrix has all entries 0. We write Z n×m or simply Z
(another common notation is 0 n×m or just 0).
1.7 Example The zero map from any three-dimensional space to any twodimensional
space is represented by the 2×3 zero matrix
( )
0 0 0
Z =
0 0 0
no matter what domain and codomain bases we use.
Exercises
̌ 1.8 Perform
(
the indicated
) (
operations,
)
if defined, or state “not defined.”
5 −1 2 2 1 4
(a)
+
6 1 1 3 0 5
( )
2 −1 −1
(b) 6 ·
1 2 3
( ) ( )
2 1 2 1
(c) +
0 3 0 3
( ) ( )
1 2 −1 4
(d) 4 + 5
3 −1 −2 1