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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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 Spring '08
 Neely
 Linear Algebra, Vector Space, 2W, Identity element

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