Objectives
- Understand the definition of a basis of a subspace.
- Recipes: basis for a column space, basis for a null space, basis of a span.
- Picture: basis of a subspace of or
- Essential vocabulary words: basis, dimension.
As we discussed in Section 3.6, a subspace is the same as a span, except we do not have a set of spanning vectors in mind. There are infinitely many choices of spanning sets for a nonzero subspace; to avoid reduncancy, usually it is most convenient to choose a spanning set with the minimal number of vectors in it. This is the idea behind the notion of a basis.
Let be a subspace of A basis of is a set of vectors in such that:
Recall that a set of vectors is linearly independent if and only if, when you remove any vector from the set, the span shrinks (Theorem 3.5.14). In other words, if is a basis of a subspace then no proper subset of will span it is a minimal spanning set.
A subspace generally has infinitely many different bases, but they all contain the same number of vectors.
We leave it as an exercise to prove that any two bases have the same number of vectors; one might want to wait until after learning the invertible matrix theorem in Section 4.5.
Let be a subspace of The number of vectors in any basis of is called the dimension of and is written
The previous example implies that any basis for has vectors in it. Let be vectors in and let be the matrix with columns
Since is a square matrix, it has a pivot in every row if and only if it has a pivot in every column. We will see in Section 4.5 that the above two conditions are equivalent to the invertibility of the matrix
Now we show how to find bases for the types of subspaces we encountered in Section 3.6, namely: a span, the column space of a matrix, and the null space of a matrix.
First we show how to compute a basis for the column space of a matrix.
The pivot columns of a matrix form a basis for
The above theorem is referring to the pivot columns in the original matrix, not its reduced row echelon form. Indeed, a matrix and its reduced row echelon form generally have different column spaces. For example, in the matrix below:
the pivot columns are the first two columns, so a basis for is
The first two columns of the reduced row echelon form certainly span a different subspace, as
but contains vectors whose last coordinate is nonzero.
Computing a basis for a span is the same as computing a basis for a column space. Indeed, the span of finitely many vectors is the column space of a matrix, namely, the matrix whose columns are
In order to compute a basis for the null space of a matrix, one has to find the parametric vector form of the solutions of the homogeneous equation
The vectors attached to the free variables in the parametric vector form of the solution set of form a basis of
In lieu of a proof, we illustrate the theorem with an example, and we leave it to the reader to generalize the argument.