Let W be a subspace of a finite-dimensional vector space V, and consider the basis {u1, u2, . . . , uk} for W. Let {u1, u2, . . . , uk, uk+1, . . . , un} be an extension of this basis to a basis for V.
(a) Prove that {uk+1 + W, uk+2 + W, . . . , un + W} is a basis for V/W.
(b) Derive a formula relating dim(V), dim(W), and dim(V/W).
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