Answer preview
To solve this, you'll need to know what set S contains and apply the fundamental theorem that any basis for R^4 must have exactly 4 linearly independent vectors. The key insight is determining how many vectors are in S and whether they're linearly independent...
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How to approach this question
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Step 1: Identify the dimension requirement - any basis for R^4 must contain exactly 4 vectors
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Step 2: Determine the size and linear independence properties of the given set S
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Step 3: Use combinatorial counting to find all 4-element subsets that are linearly independent