Linear Algebra

174 Chapter Three. Maps Between Spaces1.12 Remark In this book we use ‘linear transformation’ only in the casewhere the codomain equals the domain, but it is widely used in other texts asa general synonym for ‘homomorphism’.1.13 Example The map on R 2 that projects all vectors down to the x-axis( (x x↦→y)0)is a linear transformation.1.14 Example The derivative map d/dx: P n → P na 0 + a 1 x + · · · + a n x n d/dx↦−→ a 1 + 2a 2 x + 3a 3 x 2 + · · · + na n x n−1is a linear transformation, as this result from calculus notes: d(c 1 f + c 2 g)/dx =c 1 (df/dx) + c 2 (dg/dx).1.15 Example The matrix transpose map( ) ( )a b a c↦→c d b dis a linear transformation of M 2×2 . Note that this transformation is one-to-oneand onto, and so in fact it is an automorphism.We finish this subsection about maps by recalling that we can linearly combinemaps. For instance, for these maps from R 2 to itself( ( ) ( ( )x f 2xx g 0↦−→and ↦−→y)3x − 2yy)5xthe linear combination 5f − 2g is also a map from R 2 to itself.( ( )x 5f−2g 10x↦−→y)5x − 10y1.16 Lemma For vector spaces V and W , the set of linear functions from Vto W is itself a vector space, a subspace of the space of all functions from V toW . It is denoted L(V, W ).Proof. This set is non-empty because it contains the zero homomorphism. Soto show that it is a subspace we need only check that it is closed under linearcombinations. Let f, g : V → W be linear. Then their sum is linear(f + g)(c 1 ⃗v 1 + c 2 ⃗v 2 ) = c 1 f(⃗v 1 ) + c 2 f(⃗v 2 ) + c 1 g(⃗v 1 ) + c 2 g(⃗v 2 )and any scalar multiple is also linear.Hence L(V, W ) is a subspace.= c 1(f + g)(⃗v1 ) + c 2(f + g)(⃗v2 )(r · f)(c 1 ⃗v 1 + c 2 ⃗v 2 ) = r(c 1 f(⃗v 1 ) + c 2 f(⃗v 2 ))= c 1 (r · f)(⃗v 1 ) + c 2 (r · f)(⃗v 2 )QED