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**divinelogos** Problem:

Find a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors.

Question:

For these problems, you set up a homogeneous system of equations and find a dependency relationship between the vectors. This seems to imply that the answer is not unique, i.e., depending on what vector you solve for, you can remove that particular one from the original set. However, in the following example, my text says there is only one solution, while I think there are different ones depending on what equation you solve for.

Example:

V=P1, {2-5x,3+7x,4-x}

Now, the dependency relationship turns out to be:

−3p1(x) + p2(x) + 2p3(x) = 0