4. Let W1 and W2 be subspaces of a vector space V such that V = W1 + W2. Let B be a basis for Win W2. We extend B to a basis Bu B1 for W1, and extend B to a basis Bu B2 for W2 (where B, B1 and B2 are...

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4. Let W1 and W2 be subspaces of a vector space V such that V = W1 + W2. Let B be a basis<br>for Win W2. We extend B to a basis Bu B1 for W1, and extend B to a basis Bu B2 for W2<br>(where B, B1 and B2 are pairwise disjoint). Define W' = span B2.<br>(a) Prove that V/W1 is isomorphic to W'.<br>(b) Prove that V/W1 is isomorphic to W2/(W1n W2).<br>

Extracted text: 4. Let W1 and W2 be subspaces of a vector space V such that V = W1 + W2. Let B be a basis for Win W2. We extend B to a basis Bu B1 for W1, and extend B to a basis Bu B2 for W2 (where B, B1 and B2 are pairwise disjoint). Define W' = span B2. (a) Prove that V/W1 is isomorphic to W'. (b) Prove that V/W1 is isomorphic to W2/(W1n W2).

Jun 04, 2022

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4. Let W1 and W2 be subspaces of a vector space V such that V = W1 + W2. Let B be a basis for Win W2. We extend B to a basis Bu B1 for W1, and extend B to a basis Bu B2 for W2 (where B, B1 and B2 are...

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