Transcription of Linear Maps - University of California, Davis
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Linear Maps - University of California, Davis
MAT067 University of California, DavisWinter 2007 Linear MapsIsaiah Lankham, Bruno Nachtergaele, Anne Schilling(February 5, 2007)As we have discussed in the lecture on What is Linear Algebra? one of the main goalsof Linear algebra is the characterization of the solutions to the set ofmlinear equations innunknownsx1, .. , xna11x1+ +a1nxn= + +amnxn=bm,where all coefficientsaijandbiare inF. Linear maps and their properties that we are aboutto discuss give us a lot of insight into the characteristics of the Definition and elementary propertiesThroughout this chapterV, Ware vector spaces overF.
3. Distributive property: (T1 +T2)S = T1S+T2S and T(S1 +S2) = TS1 +TS2 where S,S1,S2 ∈ L(U,V) and T,T1,T2 ∈ L(V,W). Note that the product of linear maps is not always commutative. For example if T ∈ L(P(F),P(F)) is differentiation Tp(z) = p′(z) and S ∈ L(P(F),P(F)) is multiplication by z2 given by Sp(z) = z2p(z), then
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