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tut6 - ~x ∈ R 4 | x 1 2 x 3 = 0 and x 1-3 x 4 = 0 c The...

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Math 136 Tutorial 6 Problems 1: Determine, with proof, which of the following are subspaces of R 4 . a) The solution space of a homogeneous system with 3 equations in 4 unknowns. b) W = x 1 x 2 x 3 x 4 M (2 , 2) | x 1 + x 2 + x 3 + x 4 = 1 . c) S = { p ( x ) P 2 | p (1) = 0 } of P 2 . 2: Determine, with proof, whether the given set is a basis for P 2 . a) { x 2 + x + 1 , 2 x 2 - x + 1 , x - 2 } . b) { x 2 - 2 x + 1 , x 2 - 4 x + 4 , x + 1 , x - 2 } c) { x 2 + 1 , 3 } . 3: Invent a basis for each of the following subspaces of the given vector space. a) S = { p ( x ) P 2 | p (1) = 0 } of P 2 . b) T
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Unformatted text preview: { ~x ∈ R 4 | x 1 + 2 x 3 = 0 and x 1-3 x 4 = 0 } c) The set of all 3 × 3 diagonal matrices in M (3 , 3). 4: Let V and W be vector spaces and let T : V → W be a linear mapping with Null( T ) = { ~ } . Suppose that { ~v 1 ,...,~v k } a linearly independent set in V . Prove that { T ( ~v 1 ) ,...,T ( ~v k ) } is a linearly independent set in W . 1...
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