Linear Algebra

210 Chapter Three. Maps Between SpacesClearly( )5 0Rep B,D (5f) =5 5and so going from the matrix representing f to the matrix representing 5f justmeans multiplying all the entries by 5.We can also consider how to compute the representation of the sum of twomaps from the representation of those maps.1.2 Example Suppose that two linear maps with the same domain and codomainf, g: R 2 → R 3 are represented with respect to some bases B and D by thesematrices.Rep B,D (f) =(1 32 0)( )−2 −1Rep B,D (g) =2 4Recall the definition of the sum of two functions: if f takes ⃗v ↦→ ⃗u and g takes⃗v ↦→ ⃗w then f + g is the function that takes ⃗v ↦→ ⃗u + ⃗w. Note that where theseare the representations of the vectorsRep B (⃗v) =( )v 1v 2Rep D (⃗u) =( )u 1u 2Rep D (⃗w) =( )w 1w 2we have ⃗u + ⃗w = (u 1⃗δ 1 + u 2⃗δ 2 ) + (w 1⃗δ 1 + w 2⃗δ 2 ) = (u 1 + w 1 )⃗δ 1 + (u 2 + w 2 )⃗δ 2and so this is the representation of the vector sum.( )u 1 + w 1Rep D (⃗u + ⃗w) =u 2 + w 2Hence, since these represent the actions of f and g( ) ( ) ( ) ( ) (1 3 v 1 v 1 + 3v 2 −2 −1=2 0 v 2 2v 1 2 4this represents the action of f + g.Rep B,D (f + g) ·)v 1=v 2( ) ( )v 1 −v 1 + 2v 2=v 2 4v 1 + 4v 2( )−2v 1 − v 22v 1 + 4v 2Therefore, we compute the matrix representing the function sum by adding theentries of the two matrices representing the functions.( )−1 2Rep B,D (f + g)4 41.3 Definition The scalar multiple of a matrix is the result of entry-by-entryscalar multiplication. The sum of two same-sized matrices is their entry-by-entrysum.