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# 3x2 - R 3 so this last equation is exactly the desired(1...

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Math 260, Spring 2012 Jerry L. Kazdan Linear Maps from R 2 to R 3 As an exercise, which I hope you will (soon) realize is entirely routine, we will show that a linear map F ( X ) = Y from R 2 to R 3 must just be three linear high school equations in two variables: a 11 x 1 + a 12 x 2 = y 1 a 21 x 1 + a 22 x 2 = y 2 a 31 x 1 + a 32 x 2 = y 3 (1) Linearity means for any vectors U and V in R 2 and any scalars c F ( U + V ) = F ( U ) + F ( V ) and F ( cU ) = cF ( U ) . Idea : write X := ( x 2 , x 2 ) R 2 as X = x 1 (1 , 0) + x 2 (0 , 1) = x 1 e 1 + x 2 e 2 , where e 1 := (1 , 0) , e 2 := (0 , 1) (physicists often write e 1 as i and e 2 as j but using this notation in higher dimensions one quickly runs out of letters). Then, by the two linearity properties Y = F ( X ) = F ( x 1 e 1 + x 2 e 2 ) = F ( x 1 e 1 ) + F ( x 2 e 2 ) = x 1 F ( e 1 ) + x 2 F ( e 2 ) . But F ( e 1 ) and F ( e 2 ) are just specific vectors in
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Unformatted text preview: R 3 so this last equation is exactly the desired (1) with F ( e 1 ) = a 11 a 21 a 31 and F ( e 2 ) = a 12 a 22 a 32 . Collecting the ingredients we have found that y 1 y 2 y 3 = Y = F ( x ) = x 1 a 11 a 21 a 31 + x 2 a 12 a 22 a 32 = x 1 a 11 + x 2 a 12 x 1 a 21 + x 2 a 22 x 1 a 31 + x 2 a 32 as claimed in(1). [Last revised: January 13, 2012] 1...
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