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Linear Transformations - Mathematics

Linear TransformationsThe two basic vector operations are addition and scaling. From this perspec-tive, the nicest functions are those which preserve these operations:Def:Alinear transformationis a functionT:Rn Rmwhich satisfies:(1)T(x+y) =T(x) +T(y) for allx,y Rn(2)T(cx) =cT(x) for allx Rnandc :IfT:Rn Rmis a Linear transformation , thenT(0) = ve already met examples of Linear Transformations . Namely: ifAisanym nmatrix, then the functionT:Rn Rmwhich is matrix-vectormultiplicationT(x) =Axis a Linear transformation .(Wait: I thought matriceswerefunctions? Technically, no. matrices are lit-erally just arrays of numbers. However, matricesdefinefunctions by matrix-vector multiplication, and such functions are always Linear Transformations .)Question:Are these all the Linear Transformations there are? That is, doesevery Linear transformation come from matrix-vector multiplication? Yes:Prop :LetT:Rn Rmbe a Linear transformation .

Two Examples of Linear Transformations (1) Diagonal Matrices: A diagonal matrix is a matrix of the form D= 2 6 6 6 4 d 1 0 0 0 d 2 0. .. 0 0 0 d n 3 7 7 7 5: The linear transformation de ned by Dhas the following e ect: Vectors are...

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