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Math / Linear Algebra: A Geometric Approach 2 / Chapter 3.3 / Problem 16

Linear Algebra: A Geometric Approach | 2nd Edition | ISBN: 9781429215213 | Authors: Ted Shifrin, Malcolm Adams

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Suppose v1, . . . , vk Rn form a linearly dependent set. Prove that for some j between 1

Chapter 3, Problem 16

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QUESTION:

Suppose v1, . . . , vk Rn form a linearly dependent set. Prove that for some j between 1 and k we have vj Span (v1, . . . , vj1, vj+1, . . . , vk). That is, one of the vectors v1, . . . , vk can be written as a linear combination of the remaining vectors.

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QUESTION:

Suppose v1, . . . , vk Rn form a linearly dependent set. Prove that for some j between 1 and k we have vj Span (v1, . . . , vj1, vj+1, . . . , vk). That is, one of the vectors v1, . . . , vk can be written as a linear combination of the remaining vectors.

ANSWER:

Step 1 of 2

Prove that for some , where .

Let the set is linearly dependent. Then by definition, it satisfies the equation , where not all the are zero.

Let the nonzero constant be . From the equation , find an expression for .

Divide throughout by , since .

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