# 31 Row and Column Spaces

For every matrix , has row vectors and column vectors. Each row of a matrix is a row vector of that matrix. Similarly, each column is a column vector of that matrix. Using these vectors, we can find a basis for the row space and column space of . Let’s look at the definition for row spaces and column spaces as given by Larson and Falvo [16].

Definition VI.10
Let be an matrix.

1. The row space of is the subspace of spanned by the row vectors of .
2. The column space of is the subspace of spanned by the column vectors of

Because the row vectors of a matrix span the row space of , we can find the basis of the row space by removing any row vectors that can be written as a linear combination of the other row vectors. Similarly, the column vectors of span the column space of . So, we can find the basis of the column space by removing any column vectors that can be written as a linear combination of the other column vectors.

Before demonstrating these ideas with an example, it is helpful to have the following theorem from Larson and Falvo to make our work easier [16].

Theorem VI.11 (Basis for the Row Space of a Matrix)

If a matrix is row-equivalent to a matrix in row-echelon form, then the nonzero row vectors of form a basis for the row space of .

Now, let’s work through an example that I completed for Elementary Linear Algebra that comes from Larson and Falvo [16].

Example 65
Find a basis for the row space and a basis for the column space of the matrix

Solution
To find a basis for the row space, we first need to convert our matrix into row-echelon form using elementary row operations.

Therefore, by Theorem VI.11, a basis for the row space is

To find a basis for the column space, the method is the same but with one added step. We must first find the transpose of our matrix [16].

The transpose of an matrix is denoted as . Furthermore, the th row of is equal to the th column of , and the th column of is equal to the th row of for all and .

So, the transpose of our matrix is

Converting to row-echelon form gives us

By Theorem VI.11, a basis for the row space of the transpose of our matrix is

Because the rows of the transpose of our matrix are equal to the columns of our original matrix, the row space of the transpose of our matrix is equal to the column space of our original matrix. Therefore, a basis for the column space of our original matrix is the set of column vectors

So, we have seen row and column spaces and how to find a basis for each. Now, we will look at another type of space, the null space, and how to find a basis for this space.