Suppose V is a vector space and has a basis B = {b₁,...,bₙ}. Which of the following statements are true? (There may be more than one true choice.)
(a) Any other basis of V has exactly n vectors.
(b) V is n-dimensional.
(c) The set {b₁,...,bₙ, bₙ₊₁} for some vector bₙ₊₁ in V is linearly independent.
(d) The set {b₁,...,bₙ, bₙ₊₁} for some vector bₙ₊₁ in V doesn't span V.
(e) The Span{b₁,...,bₙ, bₙ₊₁} = V for some vector bₙ₊₁ in V.
(f) The set {b₁,...,bₙ₋₁} is linearly independent.
(g) The set {b₁,...,bₙ₋₁} spans V.
2. Suppose V is a subspace of a finite dimensional vector space W. Let B = {b₁,...,bₙ} be a basis of V. Decide whether each of the following must be true, could be true or is false. If you choose "could be true", state what requirements are necessary for it to be true.
(a) dim W ≥ n. must be T could be T F
(b) dim W ≤ dim V. must be T could be T F
(c) dim V = dim W. must be T could be T F
(d) Let x₁ ∈ W, then the set {b₁,...,bₙ, x} spans V. must be T could be T F
(e) Let x ∈ W, then the set {b₁,...,bₙ, x} spans W. must be T could be T F
(f) Let x ∈ W, then Span{b₁,...,bₙ, x} = V. must be T could be T F
(g) Let x ∈ W, then Span{b₁,...,bₙ, x} = W. must be T could be T F