Lect2-08-web

Lect2-08-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 15: Linear transformations (continued). Range and kernel. Matrix transformations. Linear mapping = linear transformation = linear function 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 . Properties of linear mappings Let L : V 1 V 2 be a linear mapping. 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 ( r 1 v 1 + r 2 v 2 ) = L ( r 1 v 1 ) + L ( r 2 v 2 ) = r 1 L ( v 1 ) + r 2 L ( v 2 ), L ( r 1 v 1 + r 2 v 2 + r 3 v 3 ) = L ( r 1 v 1 + r 2 v 2 ) + L ( r 3 v 3 ) = = r 1 L ( v 1 ) + r 2 L ( v 2 ) + r 3 L ( v 3 ), and so on. L ( 1 ) = 2 , where 1 and 2 are zero vectors in V 1 and V 2 , respectively. L ( 1 ) = L (0 1 ) = 0 L ( 1 ) = 2 . L ( v ) = L ( v ) for any v V 1 . L ( v ) = L (( 1) v ) = ( 1) L ( v ) = L ( v ). Examples of linear mappings Scaling L : V V , L ( v ) = s v , where s R . L ( x + y ) = s ( x + y ) = s x + s y = L ( x ) + L ( y ), L ( r x ) = s ( r x ) = r ( s x ) = rL ( x ). Dot product with a fixed vector : R n R , ( v ) = v v , where v R n . ( x + y ) = ( x + y ) v = x v + y v = ( x ) + ( y ), ( r x ) = ( r x ) v = r ( x v ) = r ( x ). Cross product with a fixed vector L : R 3 R 3 , L ( v ) = v v , where v R 3 . Multiplication by a fixed matrix L : R n R m , L ( v ) = A v , where A is an m n matrix and all vectors are column vectors. Linear mappings of functional vector spaces Evaluation at a fixed point : F ( R ) R , ( f ) = f ( a ), where a R . Multiplication by a fixed function L : F ( R ) F ( R ), L ( f ) = gf , where g F ( R ). Differentiation D : C 1 ( R ) C ( R ), L ( f ) = f ....
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Lect2-08-web - MATH 304, Fall 2011 Linear Algebra Homework...

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