The Column Space

The column space is the vector space spanned by the columns of A.

$\blacktriangleright$ To find a basis for the column space

1.
Leave the insertion point in the matrix.

2.
From the Matrices submenu, choose Column Basis.

$\blacktriangleright$ Matrices + Column Basis

$\left[\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right.$$\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
97 & 50 & 79 & 56 \\
49 & 63 & 57 & -59 \\
-36 & 8 & 20 & -94
\end{array}
}\right]$, column basis: $\left[\vphantom{
\begin{array}{c}
0 \\
1 \\
0 \\
0
\end{array}
}\right.$$\begin{array}{c}
0 \\
1 \\
0 \\
0
\end{array}$$\left.\vphantom{
\begin{array}{c}
0 \\
1 \\
0 \\
0
\end{array}
}\right]$,$\left[\vphantom{
\begin{array}{c}
1 \\
0 \\
0 \\
1
\end{array}
}\right.$$\begin{array}{c}
1 \\
0 \\
0 \\
1
\end{array}$$\left.\vphantom{
\begin{array}{c}
1 \\
0 \\
0 \\
1
\end{array}
}\right]$,$\left[\vphantom{
\begin{array}{c}
0 \\
0 \\
1 \\
1
\end{array}
}\right.$$\begin{array}{c}
0 \\
0 \\
1 \\
1
\end{array}$$\left.\vphantom{
\begin{array}{c}
0 \\
0 \\
1 \\
1
\end{array}
}\right]$

You can also take the transpose of A and apply Fraction-Free Gaussian Elimination to the result, because the column space of A is the row space of AT.