solution_HW4_EE441

# solution_HW4_EE441 - EE 441 Applied Linear Algebra Spring...

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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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## This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.

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solution_HW4_EE441 - EE 441 Applied Linear Algebra Spring...

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