The first subspace associated to a matrix that we’ll consider is its column space.

####
Example 3.5.7.

Consider the matrix \(A\) and its reduced row echelon form:

\begin{equation*}
A =
\left[\begin{array}{rrrrr}
2 \amp 0 \amp -4 \amp -6 \amp 0 \\
-4 \amp -1 \amp 7 \amp 11 \amp 2 \\
0 \amp -1 \amp -1 \amp -1 \amp 2 \\
\end{array}\right]
\sim
\left[\begin{array}{rrrrr}
1 \amp 0 \amp -2 \amp -3 \amp 0 \\
0 \amp 1 \amp 1 \amp 1 \amp -2 \\
0 \amp 0 \amp 0 \amp 0 \amp 0 \\
\end{array}\right],
\end{equation*}

and denote the columns of \(A\) as \(\vvec_1,\vvec_2,\ldots,\vvec_5\text{.}\)

It is certainly true that \(\col(A) =
\laspan{\vvec_1,\vvec_2,\ldots,\vvec_5}\) by the definition of the column space. However, the reduced row echelon form of the matrix shows us that the vectors are not linearly independent so \(\vvec_1,\vvec_2,\ldots,\vvec_5\) do not form a basis for \(\col(A)\text{.}\)

From the reduced row echelon form, however, we can see that

\begin{equation*}
\begin{aligned}
\vvec_3 \amp {}={} -2\vvec_1 + \vvec_2 \\
\vvec_4 \amp {}={} -3\vvec_1 + \vvec_2 \\
\vvec_5 \amp {}={} -2\vvec_2 \\
\end{aligned}\text{.}
\end{equation*}

This means that any linear combination of \(\vvec_1,\vvec_2,\ldots,\vvec_5\) can be written as a linear combination of just \(\vvec_1\) and \(\vvec_2\text{.}\) Therefore, we see that \(\col(A) = \laspan{\vvec_1,\vvec_2}\text{.}\)

Moreover, the reduced row echelon form shows that \(\vvec_1\) and \(\vvec_2\) are linearly independent, which implies that they form a basis for \(\col(A)\text{.}\) This means that \(\col(A)\) is a 2-dimensional subspace of \(\real^3\text{,}\) which is a plane in \(\real^3\text{,}\) having basis

\begin{equation*}
\threevec{2}{-4}{0},
\qquad
\threevec{0}{-1}{1}\text{.}
\end{equation*}

In general, a column without a pivot position can be written as a linear combination of the columns that have pivot positions. This means that a basis for \(\col(A)\) will always be given by the columns of \(A\) having pivot positions. This leads us to the following definition and proposition.

####
Proposition 3.5.9.

If \(A\) is an \(m\times n\) matrix, then \(\col(A)\) is a subspace of \(\real^m\) whose dimension equals \(\rank(A)\text{.}\) The columns of \(A\) that contain pivot positions form a basis for \(\col(A)\text{.}\)

For example, the rank of the matrix

\(A\) in

Example 3.5.7 is two because there are two pivot positions. A basis for

\(\col(A)\) is given by the first two columns of

\(A\) since those columns have pivot positions.

As a note of caution, we determine the pivot positions by looking at the reduced row echelon form of \(A\text{.}\) However, we form a basis of \(\col(A)\) from the columns of \(A\) rather than the columns of the reduced row echelon matrix.