Linear Algebra

Section III. Basis and Dimension 121subspaces have bases with one member, and the trivial subspace has a basiswith zero members. When we saw that diagram we could not show that theseare the only subspaces that this space has. We can show it now. The priorcorollary proves that the only subspaces of R 3 are either three-, two-, one-, orzero-dimensional. Therefore, the diagram indicates all of the subspaces. Thereare no subspaces somehow, say, between lines and planes.2.10 Corollary Any linearly independent set can be expanded to make a basis.Proof. If a linearly independent set is not already a basis then it must notspan the space. Adding to it a vector that is not in the span preserves linearindependence. Keep adding, until the resulting set does span the space, whichthe prior corollary shows will happen after only a finite number of steps. QED2.11 Corollary Any spanning set can be shrunk to a basis.Proof. Call the spanning set S. If S is empty then it is already a basis (thespace must be a trivial space). If S = {⃗0} then it can be shrunk to the emptybasis, thereby making it linearly independent, without changing its span.Otherwise, S contains a vector ⃗s 1 with ⃗s 1 ≠ ⃗0 and we can form a basisB 1 = 〈⃗s 1 〉. If [B 1 ] = [S] then we are done.If not then there is a ⃗s 2 ∈ [S] such that ⃗s 2 ∉ [B 1 ]. Let B 2 = 〈⃗s 1 , ⃗s 2 〉; if[B 2 ] = [S] then we are done.We can repeat this process until the spans are equal, which must happen inat most finitely many steps.QED2.12 Corollary In an n-dimensional space, a set of n vectors is linearly independentif and only if it spans the space.Proof. First we will show that a subset with n vectors is linearly independentif and only if it is a basis. ‘If’ is trivially true — bases are linearly independent.‘Only if’ holds because a linearly independent set can be expanded to a basis,but a basis has n elements, so this expansion is actually the set that we beganwith.To finish, we will show that any subset with n vectors spans the space if andonly if it is a basis. Again, ‘if’ is trivial. ‘Only if’ holds because any spanningset can be shrunk to a basis, but a basis has n elements and so this shrunkenset is just the one we started with.QEDThe main result of this subsection, that all of the bases in a finite-dimensionalvector space have the same number of elements, is the single most importantresult in this book because, as Example 2.9 shows, it describes what vectorspaces and subspaces there can be. We will see more in the next chapter.2.13 Remark The case of infinite-dimensional vector spaces is somewhat controversial.The statement ‘any infinite-dimensional vector space has a basis’is known to be equivalent to a statement called the Axiom of Choice (see[Blass 1984]). Mathematicians differ philosophically on whether to accept orreject this statement as an axiom on which to base mathematics (although, the