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Unformatted text preview: Partial Solution Set, Leon § 4.1 First, recall that to show L is a linear transformation, one must show that either (1) L ( x + y ) = L ( x ) + L ( y ) and L ( α x ) = αL ( x ), or (2) L ( α x + β y ) = αL ( x ) + βL ( y ), or (3) L ( α x + y ) = αL ( x ) + L ( y ). Either one of the three would suffice. 4.1.1 For each of five transformations, we are to verify linearity and describe geometrically the effect of the transformation. Verification is straightforward for all; a sketch might make it easier to see what’s going on geometrically. (a) L ( x ) = ( x 1 ,x 2 ) T . Verifying linearity is simple: we simply show that L ( α x + y ) = ( ( αx 1 + y 1 ) , ( αx 2 + y 2 )) T = ( αx 1 ,αx 2 ) T + ( y 1 ,y 2 ) T = α ( x 1 ,x 2 ) T + ( y 1 ,y 2 ) T = αL ( x ) + L ( y ) . The geometric effect is reflection across the x 2axis. (c) Reflection across the identity line x 1 = x 2 . (e) Projection onto the x 2axis. 4.1.2 Let L be the linear transformation mapping R 2 into itself defined by L ( x ) = ( x 1 cos α x 2 sin α,x 1 sin α + x 2 cos α ) T . Express x 1 ,x 2 , and L ( x ) in terms of polar coordinates. Describe geometrically the effect of the linear transformation....
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 Spring '08
 Anshelvich
 Linear Algebra, Algebra, Derivative, Vector Space

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