(Solution) - Consider the subspace of R4 given by a Determine a subset -(2025 Original AI-Free Solution)

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Consider the subspace of R4 given by

(a) Determine a subset S of the spanning set that is a basis for W.
(b) Find a basis T for W that is not a subset of the spanning set.
(c) Find the coordinate vector of

With respect to each of the bases from parts (a) and (b).