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 Leon’s 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 ℓ : R n → R , ℓ ( v ) = v · v , where v ∈ R n . ℓ ( x + y ) = ( x + y ) · v = x · v + y · v = ℓ ( x ) + ℓ ( y ), ℓ ( r x ) = ( r x ) · v = r ( x · v ) = r ℓ ( 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 ℓ : F ( R ) → R , ℓ ( 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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