Unformatted text preview: a) span { (1 ,2 , 1) , (2 , , 10) , (0 , 1 , 2) , (1 , 1 , 7) } b) The hyperplane x 1 + 2 x 2x 3 + x 4 = 0. 6. Let V and W be vector spaces and let T : V → W be a linear mapping. a) Let { ±v 1 ,...,±v k } be a nonempty linearly independent set in V . Prove that if the null space of T is { ± } then { T ( ±v 1 ) ,...,T ( ±v k ) } is also linearly independent. b) Suppose { ±v 1 ,...,±v k } is a spanning set for V and that the range of T is W . Prove that { T ( ±v 1 ) ,...,T ( ±v k ) } is a spanning set for W . c) Assume that dim V = n and dim W = n . Prove that the null space of T is { ± } if and only if the range of T is W . 7. State and prove the subspace test....
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 Winter '08
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 Linear Algebra, Addition, sample term test

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