Let V be a finite-dimensional vector space with a basis β, and let β1, . . . , βk be a partition of β(i.e., β1, β2, . . . , βk are subsets of β such that β= β1∪β2∪·· · ∪βk and βi∩βj= ∅ if i≠j). Prove that V = span(β1) ⊕span(β2) ⊕·· ·⊕span(βk).

Let {~v1, ~v2, ...,~vk} be a basis for a vector space W = sp(~v1, ~v2, ...,~vk) C R^n. Let c be a nonzero scalar. It follows that W = sp(c-v1, c-v2, ..., c~vk). Prove that the set {c-v1, c-v2, ..., c~vk} is a basis for W.

Let S be a linearly independent set of vector form a finite dimensional vector space V. Prove that there exists a basis of V containing S.

Let V be an n-dimensional vector space over F and B = {b₁,b2,...,bk} be a linearly independent set of vectors in V where n > k. Suppose W = span(B). Prove that every vector a ¤ V can be expressed as α=α₁+ α₂ where a ₁ € W and a 2 W. Show that a ₁ is unique but a 2 is not unique.