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

Chapter 1, Problem 20

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QUESTION:

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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QUESTION:

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.

ANSWER:

Step 1 of 2

Given:

Statement:a vector space is infinite-dimensional if and only if it contains an infinite linearly independent subset.

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