Linear Algebra

194 Chapter 3. Maps Between Spaces
3.III Computing Linear Maps
The prior section shows that a linear map is determined by its action on a basis.
In fact, the equation
h(�v) =h(c1 · � β1 + ···+ cn · � βn) =c1 · h( � β1)+···+ cn · h( � βn)
shows that, if we know the value of the map on the vectors in a basis, then we
can compute the value of the map on any vector �v at all just by finding the c’s
to express �v with respect to the basis.
This section gives the scheme that computes, from the representation of a
vector in the domain Rep B(�v), the representation of that vector’s image in the
codomain Rep D(h(�v)), using the representations of h( � β1), ... , h( � βn).
3.III.1 Representing Linear Maps with Matrices
1.1 Example Consider a map h with domain R 2 and codomain R 3 (fixing
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
� � � �
1 0 1
2 1
B = 〈 , 〉 and D = 〈 ⎝0⎠
, ⎝−2⎠
, ⎝0⎠〉
0 4
0 0 1
as the bases for these spaces) that is determined by this action on the vectors
in the domain’s basis.
⎛ ⎞
� � 1
2 h
↦−→ ⎝1⎠
0
1
⎛ ⎞
� � 1
1 h
↦−→ ⎝2⎠
4
0
To compute the action of this map on any vector at all from the domain, we
first express h( � β1) andh( � β2) with respect to the codomain’s basis:
and
⎛
⎝ 1
⎞ ⎛
1⎠
=0⎝
1
1
⎞
0⎠
−
0
1
⎛
⎝
2
0
⎞ ⎛
−2⎠
+1⎝
0
1
⎞
0⎠
so RepD(h( 1
� ⎛
β1)) = ⎝ 0
⎞
−1/2⎠
1
⎛
⎝ 1
⎞ ⎛
2⎠
=1⎝
0
1
⎞ ⎛
0⎠
− 1 ⎝
0
0
⎞ ⎛
−2⎠
+0⎝
0
1
⎞
0⎠
so RepD(h( 1
� ⎛
β2)) = ⎝ 1
⎞
−1⎠
0
D
D