3.3 Linear transformations - Linear transformations...

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Linear transformations Definition. Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L ( u ) in W to each u in V such that : a) L ( u + v ) = L ( u ) + L ( v ) , for every u,v V b) L ( ku ) = kL ( u ) , for every u V and every scalar k . If V = W , the linear transformation L is also called a linear operator on V . Example 1. Let L : R 3 R 2 be defined by L ( x ) =( x 1 , x 2 x 3 ) , x =( x 1 ,x 2 ,x 3 ) . y =( y 1 , y 2 , y 3 ) a). L ( x + y ) = L ( x 1 + y 1 , x 2 + y 2 , x 3 + y 3 ) = ( x 1 + y 1 , ( x 2 + y 2 ) ( x 3 + y 3 ) ) = ( x 1 + y 1 , ( x 2 x 3 ) + ( y 2 y 3 ) ) = ( x 1 , ( x 2 x 3 ) ) + ( y 1 , ( . b). L ( kx ) = L ( kx 1 ,kx 2 ,k x 3 ) =( k x 1 ,k x 2 k x 3 )= k ( x 1 , x 2 x 3 )= k L ( x ) . Thus L is linear transformation. Example 2. Let L : R 2 R 2 be defined by L ( x ) =( x 1 + 1, x 2 ) . Is L a linear transformation? a). L ( x + y ) = ( x 1 + y 1 + 1 ,x 2 + y 2 ) and L ( x ) + L ( y ) = ( x 1 + 1, x 2 ) + ( y 1 + 1, y 2 ) = ( x 1 + y 1 + 2 , x 2 + y 2 ) .
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Since L ( x + y ) ≠L ( x ) + L ( y ) , we conclude that L is not a linear transformation. Definition. The linear transformation 1). L : R 3 R 2 defined by L ( x , y, z ) =( x, y ) is called projection, 2). L : R 3 R 3 defined by L ( u ) = ru,r > 1 is called dilation, 3). L : R 3 R 3 defined by L ( u ) = ru, 0 < r < 1 is called contraction, 4). L : R 2 R 2 defined by L ( x , y ) =( x , y ) is called reflection, 5).
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