The Row Space

The row space is the vector space spanned by the row vectors of A.

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

1.
Leave the insertion point in the matrix.

2.
From the Matrices submenu, choose Row Basis.

$\blacktriangleright$ Matrices + Row 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]$, row basis:

$\displaystyle \left[\vphantom{
\begin{array}{cccc}
1 & 0 & 0 &
\dfrac{133337}{68264}
\end{array}
}\right.$$\displaystyle \begin{array}{cccc}
1 & 0 & 0 &
\dfrac{133337}{68264}
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cccc}
1 & 0 & 0 &
\dfrac{133337}{68264}
\end{array}
}\right]$,$\displaystyle \left[\vphantom{
\begin{array}{cccc}
0 & 1 & 0 & -\dfrac{74049}{34132}
\end{array}
}\right.$$\displaystyle \begin{array}{cccc}
0 & 1 & 0 & -\dfrac{74049}{34132}
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cccc}
0 & 1 & 0 & -\dfrac{74049}{34132}
\end{array}
}\right]$,$\displaystyle \left[\vphantom{
\begin{array}{cccc}
0 & 0 & 1 & -\dfrac{3085}{9752}
\end{array}
}\right.$$\displaystyle \begin{array}{cccc}
0 & 0 & 1 & -\dfrac{3085}{9752}
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cccc}
0 & 0 & 1 & -\dfrac{3085}{9752}
\end{array}
}\right]$

You can find other bases by choosing Reduced Row Echelon Form from the Matrices submenu, or by applying Fraction-Free Gaussian Elimination and then taking the nonzero rows from the result.

$\blacktriangleright$ Matrices + Reduced Row Echelon Form

$\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]$, row echelon form: $\left[\vphantom{
\begin{array}{rrrr}
1 & 0 & 0 & \dfrac{133337}{68264}\vspac...
...{c}{0} & \multicolumn{1}{c}{0} &
\multicolumn{1}{c}{0}
\end{array}
}\right.$$\begin{array}{rrrr}
1 & 0 & 0 & \dfrac{133337}{68264}\vspace{6pt} \\
0 & 1 ...
...icolumn{1}{c}{0} & \multicolumn{1}{c}{0} &
\multicolumn{1}{c}{0}
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
1 & 0 & 0 & \dfrac{133337}{68264}\vspac...
...{c}{0} & \multicolumn{1}{c}{0} &
\multicolumn{1}{c}{0}
\end{array}
}\right]$

The preceding calculation gives the same basis as found in the previous example.

$\blacktriangleright$ Matrices + Fraction-free Gaussian Elimination

$\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]$, fraction-free Gaussian elimination:

$\displaystyle \left[\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\ ...
... -1365 \\
0 & 0 & 136528 & -43190 \\
0 & 0 & 0 & 0
\end{array}
}\right.$$\displaystyle \begin{array}{rrrr}
-85 & -55 & -37 & -35 \\
0 & 1085 & -3126 & -1365 \\
0 & 0 & 136528 & -43190 \\
0 & 0 & 0 & 0
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\ ...
... -1365 \\
0 & 0 & 136528 & -43190 \\
0 & 0 & 0 & 0
\end{array}
}\right]$

 

The nonzero rows in the preceding matrix give a basis for the row space:

$\left[\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35
\end{array}
}\right.$$\begin{array}{rrrr}
-85 & -55 & -37 & -35
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
-85 & -55 & -37 & -35
\end{array}
}\right]$, $\left[\vphantom{
\begin{array}{rrrr}
0 & 1085 & -3126 & -1365
\end{array}
}\right.$$\begin{array}{rrrr}
0 & 1085 & -3126 & -1365
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
0 & 1085 & -3126 & -1365
\end{array}
}\right]$, $\left[\vphantom{
\begin{array}{rrrr}
0 & 0 & 136528 & -43190
\end{array}
}\right.$$\begin{array}{rrrr}
0 & 0 & 136528 & -43190
\end{array}$$\left.\vphantom{
\begin{array}{rrrr}
0 & 0 & 136528 & -43190
\end{array}
}\right]$.