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Linear Transformations - Stanford University

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?

Linear Transformations The two basic vector operations are addition and scaling. From this perspec-tive, the nicest functions are those which \preserve" these operations: Def: A linear transformation is a function T: Rn!Rm which satis es: (1) T(x+ y) = T(x) + T(y) for all x;y 2Rn (2) T(cx) = cT(x) for all x 2Rn and c2R.

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