Linear Algebra, 2020a

Section IV. Matrix Operations 233
1.2 Example We can do a similar exploration for the sum of two maps. Suppose
that two linear maps with the same domain and codomain f, g: R 2 → R 2 are
represented with respect to bases B and D by these matrices.
(
Rep B,D (f) =
1 3
2 0
)
( )
−2 −1
Rep B,D (g) =
2 4
Recall the definition of sum: if f does ⃗v ↦→ ⃗u and g does ⃗v ↦→ ⃗w then f + g is
the function whose action is ⃗v ↦→ ⃗u + ⃗w. Let these be the representations of the
input and output vectors.
Rep B (⃗v) =
( )
v 1
v 2
Rep D (⃗u) =
( )
u 1
u 2
Rep D (⃗w) =
( )
w 1
w 2
Where D = 〈⃗δ 1 ,⃗δ 2 〉 we have ⃗u + ⃗w = (u 1
⃗δ 1 + u 2
⃗δ 2 )+(w 1
⃗δ 1 + w 2
⃗δ 2 ) =
(u 1 + w 1 )⃗δ 1 +(u 2 + w 2 )⃗δ 2 and so this is the representation of the vector sum.
( )
u 1 + w 1
Rep D (⃗u + ⃗w) =
u 2 + w 2
Thus, since these represent the actions of of the maps f and g on the input ⃗v
( )( ) ( ) ( )( ) ( )
1 3 v 1 v 1 + 3v 2 −2 −1 v 1 −2v 1 − v 2
=
=
2 0 v 2 2v 1 2 4 v 2 2v 1 + 4v 2
adding the entries represents the action of the map f + g.
( ) ( )
v 1 −v 1 + 2v 2
Rep B,D (f + g) · =
v 2 4v 1 + 4v 2
Therefore, we compute the matrix representing the function sum by adding the
entries of the matrices representing the functions.
( )
−1 2
Rep B,D (f + g) =
4 4
1.3 Definition The scalar multiple of a matrix is the result of entry-by-entry
scalar multiplication. The sum of two same-sized matrices is their entry-by-entry
sum.
These operations extend the first chapter’s operations of addition and scalar
multiplication of vectors.
We need a result that proves these matrix operations do what the examples
suggest that they do.