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Linear Algebra | 4th Edition | ISBN: 9780130084514 | Authors: Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence
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Prove that a vector space is infinite-dimensional if and only if it contains an infinite

Chapter 1, Problem 21

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QUESTION:

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

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QUESTION:

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

ANSWER:

Step 1 of 2

Here, we need to prove that a vector space is infinite dimensional if and only if it contains an infinite linearly independent subset.

A vector space is infinite dimensional contains an infinite linearly independent subset.

Let  be the infinite dimensional.

It has a basis of infinite number of vectors  a basis has linearly independent subset of

So, contains an infinite linearly independent subset.

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