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MAT A22 Winter 2024 Problems Q1. (a) The vectors { (3, 2, 1), (2, 1, 0) } are linearly independent in R3. Find a vector v such that { (3, 2, 1), (2, 1, 0), v} is a basis of 3. Use the elimination algorithm to prove v # span ({ (3, 2, 1), (2, 1, 0) } ). (b) The vectors { (1, 2, 3), (1, 1, 2), (0, 1, 1), (1, 0, 0) } form a span- ning set in R3. Use the elimination algorithm to determine which vector can be removed to create a basis. Q2. Prove: If W C V is a infinite subspace of a vector space V then V is infinite dimensional. Q3. Suppose that V is finite dimensional. Let W1 and W2 be subspaces of V with WinW2 = {0} and W1 + W2 = W3 Prove: dim(W3) = dim(W1) + dim(W2). Q4. Find the dimension of the following subspaces. (a) W are the trace zero real 2 x 2 matrices. (b) W are the skew symmetric real 2 x 2 matrices. (c) Polynomials of degree n. Q5. Suppose we have two different subsets S, and S2 of a finite vector space V, where | Sil