Linear Algebra, 2020a

Section III. Computing Linear Maps 215
shows that this is the matrix representing h with respect to the bases.
(
)
−1/2 1 2
Rep B,D (h) =
−1/2 −1 −2
1.5 Theorem Assume that V and W are vector spaces of dimensions n and m
with bases B and D, and that h: V → W is a linear map. If h is represented by
⎛
⎞
h 1,1 h 1,2 ... h 1,n
h 2,1 h 2,2 ... h 2,n
Rep B,D (h) =
⎜
⎟
⎝ .
⎠
h m,1 h m,2 ... h m,n
B,D
B,D
and ⃗v ∈ V is represented by
⎛ ⎞
c 1
c 2...
Rep B (⃗v) =
⎜ ⎟
⎝ ⎠
c n
B
then the representation of the image of ⃗v is this.
⎛
⎞
h 1,1 c 1 + h 1,2 c 2 + ···+ h 1,n c n
h 2,1 c 1 + h 2,2 c 2 + ···+ h 2,n c n
Rep D ( h(⃗v))=
⎜
⎝
⎟
.
⎠
h m,1 c 1 + h m,2 c 2 + ···+ h m,n c n
D
Proof This formalizes Example 1.1. See Exercise 33.
QED
1.6 Definition The matrix-vector product of a m×n matrix and a n×1 vector
is this.
⎛
⎛
⎜
⎝
⎞
a 1,1 a 1,2 ... a 1,n
a 2,1 a 2,2 ... a 2,n
⎟
.
⎠
a m,1 a m,2 ... a m,n
⎞
⎛ ⎞ a 1,1 c 1 + ···+ a 1,n c n
1...
⎜ ⎟
a 2,1 c 1 + ···+ a 2,n c n
⎝c
⎠ =
⎜
⎝
⎟
. ⎠
c n
a m,1 c 1 + ···+ a m,n c n
Briefly, application of a linear map is represented by the matrix-vector
product of the map’s representative and the vector’s representative.
1.7 Remark Theorem 1.5 is not surprising, because we chose the matrix representative
in Definition 1.2 precisely to make the theorem true — if the theorem