Transcription of Linear Transformation Exercises
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Linear Transformation Exercises
Linear Transformation ExercisesOlena BormashenkoDecember 12, 20111. Determine whether the following functions are Linear transformations. Ifthey are, prove it; if not, provide a counterexample to one of the properties:(a)T:R2 R2, withT[xy]=[x+yy]Solution:ThisISa Linear Transformation . Let s check the properties:(1)T(~x+~y) =T(~x) +T(~y): Let~xand~ybe vectors inR2. Then,we can write them as~x=[x1x2], ~y=[y1y2]By definition, we have thatT(~x+~y) =T[x1+y1x2+y2]=[x1+y1+x2+y2x2+y2]andT(~x ) +T(~y) =T[x1x2]+T[y1y2]=[x1+x2x2]+[y1+y2y2]=[x1 +x2+y1+y2x2+y2]Thus, we see thatT(~x+~y) =T(~x) +T(~y), so this property holds.(2)T(c~x) =cT(~x): Let~xbe as above, and letcbe a scalar.
Linear Transformation Exercises Olena Bormashenko December 12, 2011 1. Determine whether the following functions are linear transformations. If they are, prove it; if not, provide a counterexample to one of the properties: (a) T : R2!R2, with T x y = x+ y y Solution: This IS a linear transformation. Let’s check the properties:
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