Linear Algebra

Section II. Homomorphisms 1771.14 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 shows: d(c 1 f + c 2 g)/dx =c 1 (df/dx) + c 2 (dg/dx).1.15 Example The matrix transpose operation( ) (a b a↦→c d b)cdis a linear transformation of M 2×2 . (Transpose is one-to-one and onto and so infact 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 V toW is itself a vector space, a subspace of the space of all functions from V to W.We denote the space of linear maps by 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 theoperations. Let f, g: V → W be linear. Then the sum of the two is linear(f + g)(c 1 ⃗v 1 + c 2 ⃗v 2 ) = f(c 1 ⃗v 1 + c 2 ⃗v 2 ) + 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 )= c 1(f + g)(⃗v1 ) + c 2(f + g)(⃗v2 )and any scalar multiple of a map is also linear.Hence L(V, W) is a subspace.(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 )QEDWe started this section by defining homomorphisms as a generalizationof isomorphisms, isolating the structure preservation property. Some of theproperties of isomorphisms carried over unchanged while we adapted others.