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Unformatted text preview: Solutions to Homework #2, MAT1341A Question # 1: (10 points) Let V = { [ x y ]  x, y ∈ R } . Define the operations of scalar multiplication ( ? ) and addition ( ⊕ ) on the set V as follows: for [ x 1 y 1 ], [ x 2 y 2 ] ∈ V and a ∈ R , a ? [ x 1 y 1 ] = [ ax 1 + a 1 ay 1 a + 1] [ x 1 y 1 ] ⊕ [ x 2 y 2 ] = [ x 1 + x 2 + 1 y 1 + y 2 1] , (Note that V is just the set M 1 , 2 of 1 × 2 matrices with real–number entries, but that the vector space operations defined here are not the ‘usual’ operations of addition and scalar multiplication on M 1 , 2 .) Show that the set V with these operations is a vector space; that is, check that all 10 axioms for a vector space hold in this example. In particular, you must check that the set V is closed under scalar multiplication and addition, identify the zero vector of V , and for any element [ x y ] ∈ V , identify its additive inverse [ x y ]. The remaining axioms must then be verified by evaluating both sides. Solution: So this is a set V with two operations, denoted ⊕ and ? (so that you don’t confuse them with the usual operations on M 1 , 2 ). Then by the definition of a vector space, this set V , with these operations, is a vector space if and only if all 10 axioms hold. Note that these 10 axioms are rules testing if the given (new, weird) operations behave “properly”. In other words, for each and every axiom, the operations you have to “test” are ⊕ and ? ; the usual operations are irrelevant here. So the axiom tests are: • Check closure under operations ⊕ , ? . (These are immediate, because all we need to see is that the rules for addition ⊕ and scalar multiplication ? give results which lie in M 1 , 2 , which is clearly true.) • Find the zero–vector: this amounts to solving the following equation for α , β : [ x + α + 1 y + β 1] = [ x y ] ⊕ [ α β ] = [ x y ] giving us = [ α β ] = [ 1 1]. Alternatively, one may propose the candidate = 0 ? [ x y ] = [ 1 1] but then one must verify that [ 1 1]...
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 Winter '10
 DAIGLE
 Linear Algebra, Addition, Multiplication, Scalar, Vector Space, additive inverse

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