Lect2-09-web

Lect2-09-web - MATH 304, Fall 2011 Linear Algebra Homework...

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Unformatted text preview: MATH 304, Fall 2011 Linear Algebra Homework assignment #7 (due Thursday, October 27) All problems are from Leons book. Section 4.1: 4, 7c, 7d, 17c, 19a Section 4.2: 2c, 3c, 4a, 6, 15 MATH 304 Linear Algebra Lecture 16: Matrix transformations (continued). Matrix of a linear transformation. Linear transformation Definition. Given vector spaces V 1 and V 2 , a mapping L : V 1 V 2 is linear if L ( x + y ) = L ( x ) + L ( y ), L ( r x ) = rL ( x ) for any x , y V 1 and r R . Basic properties of linear mappings: L ( r 1 v 1 + + r k v k ) = r 1 L ( v 1 ) + + r k L ( v k ) for all k 1, v 1 , . . . , v k V 1 , and r 1 , . . . , r k R . L ( 1 ) = 2 , where 1 and 2 are zero vectors in V 1 and V 2 , respectively. L ( v ) = L ( v ) for any v V 1 . Matrix transformations Any m n matrix A gives rise to a transformation L : R n R m given by L ( x ) = A x , where x R n and L ( x ) R m are regarded as column vectors. This transformation is linear . Example. L x y z = 1 0 2 3 4 7 0 5 8 x y z . Let e 1 = (1 , , 0), e 2 = (0 , 1 , 0), e 3 = (0 , , 1) be the standard basis for R 3 . We have that L ( e 1 ) = (1 , 3 , 0), L ( e 2 ) = (0 , 4 , 5), L ( e 3 ) = (2 , 7 , 8). Thus L ( e 1 ) , L ( e 2 ) , L ( e 3 ) are columns of the matrix. Problem. Find a linear mapping L : R 3 R 2 such that L ( e 1 ) = (1 , 1), L ( e 2 ) = (0 , 2), L ( e 3 ) = (3 , 0), where e 1 , e 2 , e 3 is the standard basis for R 3 . L ( x , y , z ) = L ( x e 1 + y e 2 + z e 3 ) = xL ( e 1 ) + yL ( e 2 ) + zL ( e 3 ) = x (1 , 1) + y (0 , 2) + z (3 , 0) = ( x + 3 z , x 2 y ) L ( x , y , z ) = parenleftbigg x + 3 z x 2 y parenrightbigg = parenleftbigg 1 0 3 1 2 0 parenrightbigg x y z Columns of the matrix are vectors L ( e 1 ) , L ( e 2 ) , L ( e 3 ). Theorem Suppose L : R n R m is a linear map. Then there exists an m n matrix A such that L ( x ) = A x for all x R n . Columns of A are vectors L ( e 1 ) , L ( e 2 ) , . . . , L ( e n ), where e 1 , e 2 , . . . , e n is the standard basis for R n . y = A x y 1 y 2 ....
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Lect2-09-web - MATH 304, Fall 2011 Linear Algebra Homework...

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