Linear Algebra, 2020a

214 Chapter Three. Maps Between Spaces
1.2 Definition Suppose 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 1,1
h 1,n
h 2,1
h 2,n
Rep D (h(⃗β 1 )) =
⎜ ⎟ ... Rep D (h(⃗β n )) =
⎜ ⎟
⎝ . ⎠
⎝ . ⎠
h m,1 h m,n
then
⎛
Rep B,D (h) =
⎜
⎝
D
⎞
h 1,1 h 1,2 ... h 1,n
h 2,1 h 2,2 ... h 2,n
⎟
.
⎠
h m,1 h m,2 ... h m,n
is the matrix representation of h with respect to B, D.
In that matrix the number of columns n is the dimension of the map’s domain
while the number of rows m is the dimension of the codomain.
1.3 Remark As with the notation for represenation of a vector, the Rep B,D
notation here is not standard. The most common alternative is [h] B,D .
We use lower case letters for a map, upper case for the matrix, and lower case
again for the entries of the matrix. Thus for the map h, the matrix representing
it is H, with entries h i,j .
1.4 Example If h: R 3 → P 1 is
⎛
then where
⎜
a ⎞
1
⎟
⎝a 2 ⎠
a 3
h
↦−→ (2a 1 + a 2 )+(−a 3 )x
B,D
⎛
⎜
0
⎞ ⎛
⎟ ⎜
0
⎞ ⎛ ⎞
2
⎟ ⎜ ⎟
B = 〈 ⎝0⎠ , ⎝2⎠ , ⎝0⎠〉 D = 〈1 + x, −1 + x〉
1 0 0
the action of h on B is this.
⎛
⎜
0
⎞
⎟
⎝0⎠
↦−→ h
−x
1
A simple calculation
( )
−1/2
Rep D (−x) =
−1/2
D
⎛
⎜
0
⎞
⎟
⎝2⎠
↦−→ h
2
0
( )
1
Rep D (2) =
−1
⎛
⎜
2
⎞
⎟
⎝0⎠
↦−→ h
4
0
D
D
( )
2
Rep D (4) =
−2
D