Linear Algebra, 2020a

224 Chapter Three. Maps Between Spaces
2.1 Example Consider this matrix.
⎛ ⎞
1 2
⎜ ⎟
H = ⎝3 4⎠
5 6
It is 3×2 so any map that it defines must carry a dimension 2 domain to a
dimension 3 codomain. We can choose the domain and codomain to be R 2 and
P 2 , with these bases.
( ) (
B = 〈 ,
1
1
1
−1
)
〉 D = 〈x 2 ,x 2 + x, x 2 + x + 1〉
Then let h: R 2 → P 2 be the function defined by H. We will compute the image
under h of this member of the domain.
( )
−3
⃗v =
2
The computation is straightforward.
⎛
⎜
1 2
⎞
⎛
( )
⎟ −1/2 ⎜
−11/2
⎞
⎟
Rep D (h(⃗v)) = H · Rep B (⃗v) = ⎝3 4⎠
= ⎝−23/2⎠
−5/2
5 6
−35/2
From its representation, computation of h(⃗v) is routine (−11/2)(x 2 )−(23/2)(x 2 +
x)−(35/2)(x 2 + x + 1) =(−69/2)x 2 −(58/2)x −(35/2).
2.2 Theorem Any matrix represents a homomorphism between vector spaces of
appropriate dimensions, with respect to any pair of bases.
Proof We must check that for any matrix H and any domain and codomain
bases B, D, the defined map h is linear. If ⃗v, ⃗u ∈ V are such that
⎛ ⎞
⎛ ⎞
v 1
u 1
⎜ .
Rep B (⃗v) = ⎝
⎟
⎜ .
. ⎠ Rep B (⃗u) = ⎝
⎟
. ⎠
v n u n
and c, d ∈ R then the calculation
h(c⃗v + d⃗u) = ( h 1,1 (cv 1 + du 1 )+···+ h 1,n (cv n + du n ) ) · ⃗δ 1 +
···+ ( h m,1 (cv 1 + du 1 )+···+ h m,n (cv n + du n ) ) · ⃗δ m
= c · h(⃗v)+d · h(⃗u)
supplies that check.
QED