Linear Algebra

200 Chapter 3. Maps Between Spaces
�v
t θ
↦−→
tθ(�v)
we start by fixing bases. Using E2 both as a domain basis and as a codomain
basis is natural, Now, we find the image under the map of each vector in the
domain’s basis.
� �
1 tθ
↦−→
0
� cos θ
sin θ
� � 0
1
�
tθ
↦−→
� �
− sin θ
cos θ
Then we represent these images with respect to the codomain’s basis. Because
this basis is E2, vectors are represented by themselves. Finally, adjoining the
representations gives the matrix representing the map.
� �
cos θ − sin θ
RepE2,E2 (tθ) =
sin θ cos θ
The advantage of this scheme is that just by knowing how to represent the image
of the two basis vectors, we get a formula that tells us the image of any vector
at all; here a vector rotated by θ = π/6.
� �
3 tπ/6
↦−→
−2
�√
3/2 −1/2
1/2 √ �� �
3
3/2 −2
≈
� �
3.598
−0.232
(Again, we are using the fact that, with respect to E2, vectors represent themselves.)
We have already seen the addition and scalar multiplication operations of
matrices and the dot product operation of vectors. Matrix-vector multiplication
is a new operation in the arithmetic of vectors and matrices. Nothing in Definition
1.5 requires us to view it in terms of representations. We can get some
insight into this operation by turning away from what is being represented, and
instead focusing on how the entries combine.
1.9 Example In the definition the width of the matrix equals the height of
the vector. Hence, the first product below is defined while the second is not.
� �
1 0 0
4 3 1
⎛
⎝ 1
⎞
� � � �� �
0⎠
1 1 0 0 1
=
6 4 3 1 0
2
One reason that this product is not defined is purely formal: the definition requires
that the sizes match, and these sizes don’t match. (Behind the formality,
though, we have a reason why it is left undefined—the matrix represents a map
with a three-dimensional domain while the vector represents a member of a
two-dimensional space.)