Linear Algebra

Section IV. Matrix Operations 209IVMatrix OperationsThe prior section shows how matrices represent linear maps. When we see a newidea, a good strategy is to explore how it interacts with things that we alreadyunderstand. In the first subsection below we will see how the representationof a scalar product r · f relates to the representation of f, and also how therepresentation of the sum of two maps f + g relates to the representations of fand g. In the later subsections we will explore the representation of linear mapcomposition and inverse.IV.1Sums and Scalar ProductsWe start with an example showing the relationship between the representationof a function and the representation of a scalar multiple of that function.1.1 Example Let f: V → W be a linear function represented with respect to somebases by this matrix.( )1 0Rep B,D (f) =1 1Consider the scalar multiple map 5f: V → W. We want to see how to computeRep B,D (5f) from Rep B,D (f).The difference between the functions is that if f takes ⃗v ↦→ ⃗w then 5f takes⃗v ↦→ 5⃗w. So consider the representations of the domain and codomain vectorsthat are associated by f.Rep B (⃗v) =( )v 1v 2Rep D (⃗w) =( )w 1w 2The representation above says that ⃗w = w 1⃗δ 1 + w 2⃗δ 2 (where the basis D is〈⃗δ 1 ,⃗δ 2 〉). Since 5⃗w = 5 · (w 1⃗δ 1 + w 2⃗δ 2 ) = (5w 1 )⃗δ 1 + (5w 2 )⃗δ 2 we have that 5fassociates ⃗v with the vector having this representation.Rep D (5⃗w) =( )5w 15w 2So, changing the map from f to 5f has the effect of changing the representationof the codomain vector by multiplying its entries by 5.That gives us the relationship between the representation of the action of fand the representation of the action of 5f.( ) ( ) ( )1 0 v 1 v 1=1 1 v 2 v 1 + v 2Rep B,D (5f) ·( ) ( )v 1 5v 1=v 2 5v 1 + 5v 2