Do Row Operations Change the Column Space of a Matrix?
Row operations can change the column space of a matrix. But why might they do so, and do row operations always change the column space? Let’s recall a few definitions before we explore these questions.
The Column Space of a Matrix
Recall that the column space of a matrix is the subspace spanned by the columns of the matrix.
Let’s denote the column space of a matrix as Col($A$). If $A$ has $n$ rows, then the columns of $A$ have $n$ entries, and Col$(A)$ is a subspace of $\mathbb R^n$.
Row Operations
To explore how row operations can change the column space of a matrix, we also need to consider the three elementary row operations below.
- add a non-zero multiple of a row to another row
- multiply a row by a non-zero number
- swap two rows
Now that we are on the same page, let’s return to our questions by going over a few examples.
Examples
Consider the matrices:
$$A = \begin{pmatrix} 0 & 0 \\ 0 & 2 \end{pmatrix} , \qquad B = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} , \qquad E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$$
Swapping the last two rows of $A$ is a row operation that gives us a matrix in echelon form, $E$. Their column spaces are sketched below.
Notice how the column spaces of $A$ and $B$ are the same. Scaling the first row of $A$ did not change its column space. So row operations do not always affect Col$(A)$. But they can.
Col$(A)$ and Col$(E)$ are different because the row swap transformed the column space of $A$ from the span of $\begin{pmatrix} 0 \\ 1 \end{pmatrix}$ to the span of $\begin{pmatrix} 1 \\ 0 \end{pmatrix}$.
In general, the the column space of a matrix can change with row operations, but they don’t have to.