Linear Algebra, 2020a

Section III. Computing Linear Maps 223
III.2
Any Matrix Represents a Linear Map
The prior subsection shows that the action of a linear map h is described by a
matrix H, with respect to appropriate bases, in this way.
⎛ ⎞
⎛
⎞
v 1
h 1,1 v 1 + ···+ h 1,n v n
⎜ ⎟ h
⎜
⎟
⃗v = ⎝ . ⎠ ↦−→ h(⃗v) = ⎝
H
. ⎠ (∗)
v n h m,1 v 1 + ···+ h m,n v n
B
Here we will show the converse, that each matrix represents a linear map.
So we start with a matrix
⎛
⎞
h 1,1 h 1,2 ... h 1,n
h 2,1 h 2,2 ... h 2,n
H =
⎜
⎝
⎟
.
⎠
h m,1 h m,2 ... h m,n
and we will describe how it defines a map h. We require that the map be
represented by the matrix so first note that in (∗) the dimension of the map’s
domain is the number of columns n of the matrix and the dimension of the
codomain is the number of rows m. Thus, for h’s domain fix an n-dimensional
vector space V and for the codomain fix an m-dimensional space W. Alsofix
bases B = 〈⃗β 1 ,...,⃗β n 〉 and D = 〈⃗δ 1 ,...,⃗δ m 〉 for those spaces.
Now let h: V → W be: where ⃗v in the domain has the representation
⎛
Rep B (⃗v) =
⎜
⎝
⎞
v 1
. ⎟
. ⎠
v n
then its image h(⃗v) is the member of the codomain with this representation.
⎛
⎞
h 1,1 v 1 + ···+ h 1,n v n
⎜
⎟
Rep D ( h(⃗v))= ⎝ . ⎠
h m,1 v 1 + ···+ h m,n v n
That is, to compute the action of h on any ⃗v ∈ V, first express ⃗v with respect to
the basis ⃗v = v 1
⃗β 1 + ···+ v n
⃗β n and then h(⃗v) =(h 1,1 v 1 + ···+ h 1,n v n ) · ⃗δ 1 +
···+(h m,1 v 1 + ···+ h m,n v n ) · ⃗δ m .
Above we have made some choices; for instance V canbeanyn-dimensional
space and B could be any basis for V, soH does not define a unique function.
However, note once we have fixed V, B, W, and D then h is well-defined since ⃗v
has a unique representation with respect to the basis B and the calculation of ⃗w
from its representation is also uniquely determined.
B
D
D