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Prove the following generalization of the replacement theorem. Let ft be a basis for a

Chapter 1, Problem 7

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QUESTION:

Prove the following generalization of the replacement theorem. Let ft be a basis for a vector space V, and let S be a linearly independent subset of V. There exists a subset Si of ft such that S U Si is a basis forV.

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QUESTION:

Prove the following generalization of the replacement theorem. Let ft be a basis for a vector space V, and let S be a linearly independent subset of V. There exists a subset Si of ft such that S U Si is a basis forV.

ANSWER:

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Statement:

Prove the following generalization of the replacement theorem. Let  be a basis for a vector space V, and let S be a linearly independent subset of V. There exists a subset  of  such that  is a basis for V.

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