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**Unformatted text preview: **MAS 4105—Practice Problems for Exam #1 1. Find a basis β for each vector space V . (No justification needed for this problem.) (a) V = M 2 × 2 ( F ) (b) V = { ( x 1 ,x 2 ,x 3 ) ∈ R 3 : x 1 + x 2 − 5 x 3 = 0 } (c) V = { f ∈ P 2 ( F ) : f (0) = 0 } 2. In each case determine whether W is a subspace of R 3 . If W is a subspace check that the subspace conditions are satisfied. If W is not a subspace show that W violates one of these conditions. (a) W = { ( x 1 ,x 2 ,x 3 ) ∈ R 3 : x 1 − 5 x 2 = 0 } (b) W = { ( x 1 ,x 2 ,x 3 ) ∈ R 3 : x 2 ≥ } 3. (a) Determine whether the set S = { (1 , 1 , 2) , (0 , 1 , 1) , (1 , 1 , 3) } is a basis for R 3 . Explain your reasoning, and be sure to use Gaussian elimination if you need to solve a system of linear equations. (b) Let S = { vectorx 1 ,vectorx 2 ,...,vectorx n } be a subset of V . Prove that if vector y ∈ Span( S ) then the set S ∪ { vector y } = { vectorx 1 ,vectorx 2 ,...,vectorx n ,vector y } is linearly dependent....

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