Rank and Dimension

The rank of a matrix is the dimension of the column space. It is the same as the dimension of the row space or the number of nonzero singular values.

$\blacktriangleright$ Matrices + Rank

$\left[\vphantom{
\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}
}\right.$$\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}
}\right]$, rank: 2

$\blacktriangleright$ Matrices + Row Basis

$\left[\vphantom{
\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}
}\right.$$\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}
}\right]$, row basis: $\left\{\vphantom{ \left[
\begin{array}{cccc}
1 & 0 & -16 & -3
\end{array}
...
...y}{cccc}
0 & 1 &
\frac{121}{5} & \frac{26}{5}
\end{array}
\right] }\right.$$\left[\vphantom{
\begin{array}{cccc}
1 & 0 & -16 & -3
\end{array}
}\right.$$\begin{array}{cccc}
1 & 0 & -16 & -3
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
1 & 0 & -16 & -3
\end{array}
}\right]$,$\left[\vphantom{
\begin{array}{cccc}
0 & 1 &
\frac{121}{5} & \frac{26}{5}
\end{array}
}\right.$$\begin{array}{cccc}
0 & 1 &
\frac{121}{5} & \frac{26}{5}
\end{array}$$\left.\vphantom{
\begin{array}{cccc}
0 & 1 &
\frac{121}{5} & \frac{26}{5}
\end{array}
}\right]$$\left.\vphantom{ \left[
\begin{array}{cccc}
1 & 0 & -16 & -3
\end{array}
\...
...}{cccc}
0 & 1 &
\frac{121}{5} & \frac{26}{5}
\end{array}
\right] }\right\}$

$\blacktriangleright$ Matrices + Column Basis

$\left[\vphantom{
\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}
}\right.$$\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-8 & -5 & 7 & -2 \\
7 & 5 & 9 & 5 \\
1 & 0 & -16 & -3 \\
8 & 5 & -7 & 2
\end{array}
}\right]$, column basis: $\left[\vphantom{
\begin{array}{r}
0 \\
1 \\
-1 \\
0
\end{array}
}\right.$$\begin{array}{r}
0 \\
1 \\
-1 \\
0
\end{array}$$\left.\vphantom{
\begin{array}{r}
0 \\
1 \\
-1 \\
0
\end{array}
}\right]$,$\left[\vphantom{
\begin{array}{r}
1 \\
0 \\
-1 \\
-1
\end{array}
}\right.$$\begin{array}{r}
1 \\
0 \\
-1 \\
-1
\end{array}$$\left.\vphantom{
\begin{array}{r}
1 \\
0 \\
-1 \\
-1
\end{array}
}\right]$