Linear Algebra

120 Chapter Two. Vector Spacesthat any other basis D = 〈 ⃗ δ 1 , ⃗ δ 2 , . . .〉 also has the same number of members, n.Because B has minimal size, D has no fewer than n vectors. We will argue thatit cannot have more than n vectors.The basis B spans the space and ⃗ δ 1 is in the space, so ⃗ δ 1 is a nontrivial linearcombination of elements of B. By the Exchange Lemma, ⃗ δ 1 can be swapped fora vector from B, resulting in a basis B 1 , where one element is ⃗ δ and all of then − 1 other elements are β’s. ⃗The prior paragraph forms the basis step for an induction argument. Theinductive step starts with a basis B k (for 1 ≤ k < n) containing k members of Dand n − k members of B. We know that D has at least n members so there is a⃗ δk+1 . Represent it as a linear combination of elements of B k . The key point: inthat representation, at least one of the nonzero scalars must be associated witha β ⃗ i or else that representation would be a nontrivial linear relationship amongelements of the linearly independent set D. Exchange ⃗ δ k+1 for β ⃗ i to get a newbasis B k+1 with one ⃗ δ more and one β ⃗ fewer than the previous basis B k .Repeat the inductive step until no β’s ⃗ remain, so that B n contains ⃗ δ 1 , . . . , ⃗ δ n .Now, D cannot have more than these n vectors because any ⃗ δ n+1 that remainswould be in the span of B n (since it is a basis) and hence would be a linear combinationof the other ⃗ δ’s, contradicting that D is linearly independent. QED2.4 Definition The dimension of a vector space is the number of vectors inany of its bases.2.5 Example Any basis for R n has n vectors since the standard basis E n hasn vectors. Thus, this definition generalizes the most familiar use of term, thatR n is n-dimensional.2.6 Example The space P n of polynomials of degree at most n has dimensionn+1. We can show this by exhibiting any basis — 〈1, x, . . . , x n 〉 comes to mind —and counting its members.2.7 Example A trivial space is zero-dimensional since its basis is empty.Again, although we sometimes say ‘finite-dimensional’ as a reminder, in therest of this book all vector spaces are assumed to be finite-dimensional. Aninstance of this is that in the next result the word ‘space’ should be taken tomean ‘finite-dimensional vector space’.2.8 Corollary No linearly independent set can have a size greater than thedimension of the enclosing space.Proof. Inspection of the above proof shows that it never uses that D spans thespace, only that D is linearly independent.QED2.9 Example Recall the subspace diagram from the prior section showing thesubspaces of R 3 . Each subspace shown is described with a minimal spanningset, for which we now have the term ‘basis’. The whole space has a basis withthree members, the plane subspaces have bases with two members, the line