Let V be a vector space having dimension n, and let S be a subset of V that generates V.(a)  Prove that there is a subset of S that is a basis for V. (Be careful not to assume that S is finite.)(b) Prove that S contains at least n vectors.

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Let V be a vector space having dimension n, and let S be a subset of V that generates V.

(a)  Prove that there is a subset of S that is a basis for V. (Be careful not to assume that S is finite.)

(b) Prove that S contains at least n vectors.

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Step 1

(a) If we can find a subset containing n elements in S such that they are linearly independent, then the subset is a basis for V since the dimension of V is n.

Step 2

Assume there is no subset of S which is a basis for V . Then we can never find n elements which are linearly independent, i.e. for each subset T of S, if T is linearly independent, then the number of elements in T, denoted by |T|, must not equal to n, which means |T| < n.

Step 3

We can find a linearly independent subset W such that |W| ≥ |T| for each linearly independent subset T. From above argument, |W| < n. If there exists one element v in S such that W ∪ {v} is linearly independent, then W ∪ {v} is a linearly independent subset of S such that |W ∪ {v}|...

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