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Suppose we have a vector space V of dimension n. Let R be a linearly independent...

Suppose we have a vector space V of dimension n. Let R be a linearly independent set with order n−2. Let S be a spanning set with order n+ 2. Outline a strategy to extend R to a basis for V. Outline a strategy to pare down S to a basis for V .

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R is a linearly independent set with order n-2 and S is a spanning set with order n+2. To extend R to a basis for V, we will try to express the vectors in S as linear combinations of the vectors in R. The vectors in S which cannot be expressed as linear combinations of the vectors in R may be added to R to create a a linearly independent set with order > n-2. If we are able to locate 2 such vectors then the order of the larger set created this way will be n. This set will form a basis for V.

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