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081st_tut4sol - MATH1111/2008-09/Tutorial IV Solution 1...

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MATH1111/2008-09/Tutorial IV Solution 1 Tutorial IV Suggested Solution 1. Let V be a vector space and x , y V . Complete the following according to the definition of vector spaces. (a) A student says that it is ambiguous to write - 2 x , because one may interpret - 2 x in the different ways - (2 x ) , ( - 1)(2 x ) , ( - 2) x , 2( - x ) , · · · What is your comment? Give your justification. (b) Prove x + x = 2 x . (c) How is subtraction in V defined? (i.e. What is x - y ?) Ans . (a) No, it is because all different interpretations are equal. For instance, - (2 x ) is the element satisfying 2 x + ( - (2 x ) ) = 0 (the zero vector in V ). So - (2 x ) = ( - 1)(2 x ) by Theorem 3.1.1 (iii). For your own good, prove that it is equal to all other interpretations. (b) x + x = 1 · x + 1 · x by (A8) = (1 + 1) x by (A6) = 2 x . (c) x - y = x + ( - y ).
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MATH1111/2008-09/Tutorial IV Solution 2 2. Are the following sets vector spaces with the indicated operationis? If not, why not? (a) The set V of all polynomials of degree 3 together with 0; operations are the usual operations on polynomials.
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