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# A8S - ASSIGNMENT 8 Solutions Let V be the vector space of 3...

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ASSIGNMENT 8 Solutions Let V be the vector space of 3 × 3 matrices. 1. [4 marks] Give an example of a vector v V , and bases B and C for a subspace U V , such that [ v ] B = 0 1 0 and [ v ] C = 1 0 - 2 . The number of possible answers is uncountably infinite. Note that B and C will each have three elements. Take, for example, B = 1 0 0 0 0 0 0 0 0 , 1 0 - 2 0 0 0 0 0 0 , 0 0 1 0 0 0 0 0 0 and C = 1 0 0 0 0 0 0 0 0 , 0 1 0 0 0 0 0 0 0 , 0 0 1 0 0 0 0 0 0 . That C is linearly independent is obvious; that B is linearly independent is easily shown. Then let v = 1 0 - 2 0 0 0 0 0 0 . 2. [3 marks] Find P C B and P B C for your bases B and C from the previous question. Clearly we have P C B = 1 1 0 0 0 1 0 - 2 0 . Inverting this yields P B C = 1 0 1 / 2 0 0 - 1 / 2 0 1 0 . Let u and v be vectors in R n . 3. [3 marks] Suppose neither u , nor v is equal to 0 , and let u = u u and v = v v .

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A8S - ASSIGNMENT 8 Solutions Let V be the vector space of 3...

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