MAT224H1 Midterm: MAT224 MIDTERM SELF GENERATED SOLUTION 2008.pdf

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[4] 1(a) the additive inverse -v of a vector v in a vector space is unique. Let v1 and v2 be two additve inverses for v. then, on one hand, since v1 is an additive inverse for v v + v1 + v2 = (v + v1) + v2 = 0 + v2 = v2. On the other hand, since v2 is an additive inverse for v true false v + v1 + v2 = v + v2 + v1 = (v + v2) + v1 = 0 + v1 = v1. This gives v1 = v2 and the additive inverse is unique. [4] 1(b) let f be an isomorphism from a vector space v to a vector space w . If f (v) = 0, then false v = 0. true. If f is an isomorphism then it is a one-to-one, onto linear transformation. Since f is a linear transformation, it maps 0v to 0w .

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