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practice-test1 - MAS 4105 PRACTICE MID-TERM EXAM SPRING...

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MAS 4105 — PRACTICE MID-TERM EXAM — SPRING 2011 NAME GROUP NUMBER Instructions: Write in complete sentences. All work should be written in a proper and coherent manner. Each member of a team must submit the solution of a different problem. Do only SIX problems. If you do more than six problems, then the best 6 problems are counted. TOTAL POSSIBLE : 60 points. (1) [5 + 5 = 10 points ] Let V be a vector space over a field F and let 0 denote the zero vector. (i) Prove that a 0 = 0 for each a F . (ii) Prove that 0 ⃗x = 0 for each ⃗x V . (2) [2 + 2 + 6 = 10 points ] (i) Define what it means for a subset S of a vector space to be linearly dependent. (ii) Let S be a nonempty subset of a vector space V . Define what it means for a vector ⃗v V to be a linear combination of vectors in S . Define the span of S which is denoted by span( S ). (iii) Prove that S = { x 2 + 2 x + 1 , x 2 + 3 , x 2 } is a linearly independent subset of P 2 ( R ). Explain why span( S ) = P 2 ( R ).
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