tut3 - Math 235 Tutorial 3 Problems 1: Define an explicit...

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Unformatted text preview: Math 235 Tutorial 3 Problems 1: Define an explicit isomorphism from the vector space S = 0 x1 | x1 + x2 = x3 x2 x3 to T = {p(x) ∈ P2 | p(1) = 0} to establish that the spaces are isomorphic. Prove that your map is an isomorphism. 2: Prove that if L : V → W and M : W → U are both onto linear mappings then M ◦ L is onto. 3: Let V and W be n-dimensional vector spaces with bases B and C respectively. Let L : V → W be a linear mapping. Prove that L is an isomorphism if and only if the matrix of L with respect to B and C has rank n. (Prove this without using the fact that the rank of L equals the rank of any matrix of L.) 1 ...
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