Linear Algebra, 2020a

Section III. Computing Linear Maps 217
We can illustrate Theorem 1.5 by computing the matrix-vector product representing
this action by the projection map.
⎛
⎜
2
⎞
( )
⎟ 2
π( ⎝2⎠) =
2
1
Represent the domain vector with respect to the domain’s basis
⎛ ⎞ ⎛
2
⎜ ⎟ ⎜
1
⎞
⎟
Rep B ( ⎝2⎠) = ⎝2⎠
1 1
to get this matrix-vector product.
⎛
⎜
2
⎞
(
)
⎟ 1 0 −1
Rep D ( π( ⎝2⎠))=
−1 1 1
1
B
B,D
⎛
⎜
1
⎞
⎟
⎝2⎠
1
Expanding this into a linear combination of vectors from D
( ) ( ) ( )
2 1 2
0 · + 2 · =
1 1 2
B
( )
0
=
2
checks that the map’s action is indeed reflected in the operation of the matrix.
We will sometimes compress these three displayed equations into one.
⎛
⎜
2
⎞ ⎛
⎟ ⎜
1
⎞
( ) ( )
⎟ h 0 2
⎝2⎠ = ⎝2⎠
↦−→ =
H 2 2
1 1
D
B
We now have two ways to compute the effect of projection, the straightforward
formula that drops each three-tall vector’s third component to make a two-tall
vector, and the above formula that uses representations and matrix-vector
multiplication. The second way may seem complicated compared to the first,
but it has advantages. The next example shows that for some maps this new
scheme simplifies the formula.
1.10 Example To represent a rotation map t θ : R 2 → R 2 that turns all vectors in
the plane counterclockwise through an angle θ
D
⃗u
t π/6
−→
t π/6 (⃗u)