solution_HW4_EE441

solution_HW4_EE441 - EE 441 - Applied Linear Algebra,...

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Unformatted text preview: EE 441 - Applied Linear Algebra, Spring 2012 Homework 4 Solutions 1. For the given triple ( R n , , ), we verify the axioms of vector space: Not associative since x ( y z ) = x ( y- z ) = x- y + z ( x y ) z = ( x- y ) z = x- y- z No identity element (there is a left identity, but no right identity element) since x =- x and x 0 = x No inverse element (since there is no identity). Not abelian since: x y = x- y and y x = y- x 1 x =- x 6 = x . ( c 1 c 2 ) x =- c 1 c 2 x while c 1 ( c 2 x ) = c 1 (- c 2 x ) = c 1 c 2 x c ( x + y ) =- c ( x + y ) =- c x- c y = c x + c y This axiom holds! ( c 1 + c 2 ) x =- ( c 1 + c 2 ) x =- c 1 x- c 2 x = c 1 x + c 2 x This axiom also holds. 1 The only axioms that hold are the distributive properties of scalar multi- plication. 2. To check if (3 ,- 1 , ,- 1) is in the span of the other vectors, we need to find real numbers a,b,c such that: a 2- 1 3 2 + b - 1 1 1- 3 + c...
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solution_HW4_EE441 - EE 441 - Applied Linear Algebra,...

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