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# assign6 - Math 136 Assignment 6 Due Wednesday Mar 3rd 1...

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Math 136 Assignment 6 Due: Wednesday, Mar 3rd 1. Determine, with proof, which of the following are subspaces of the given vector space. Find a basis for each subspace. a) A = {(1':l,X2,X:3) E]R3 12~rl +X3 = O,Xl +X2 -X3 = o} ofIR 3 . b) B = {ax2 + b~r + c E P2 I b 2 - 4ac = o} of P2· c) C = {ax 2 + b:r + e E P2 I a - c = b} of P 2 · d) D = {[~ ~] E M(2, 2) I ad - be = o} of AI(2, 2). e) E = { [~ ~] E Al (2, 2) I a - b + e = ° } of ,\1 (2, 2). 2. Determine, with proof, whether the given set is a basis for P 2 · a) {I + 5x + x 2 , 2 + :C:, 3 + 2 x + X2) } . b) {I + 2x + 3x2, 3 + 2x + x 2 , 5 + 6x + 7 x 2 }. 3. Invent a basis for each of the following subspaces of A1(2, 2). a) The set of all 2 x 2 lower triangular matrices. b) The set of all 2 x 2 diagonal matrices. 4. Let S = {VI, ... , 'Un} be a set of vectors in a vector space 1/'. Prove that span S is a subspace of 1/. 5. Let S = {(a, b) E ]R2 I b > o} and define addition by (a, b) + (c, d) = (ad + be, bd) and define scalar multiplication by k(a, b) = (kab k - l , b k ). Prove that S is a vector space over lR. 6. Let 1I and TV be vector spaces and let T : F -+ n y be a linear mapping.

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assign6 - Math 136 Assignment 6 Due Wednesday Mar 3rd 1...

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