Basis
A basis of the null space of a matrix may be computed by Gaussian elimination.
For this purpose, given an m × n matrix A, we construct first the row augmented matrix where I is the n × n identity matrix.
Computing its column echelon form by Gaussian elimination (or any other available method), we get a matrix A basis of the null space of A consists in the non zero columns of C such that the corresponding column of B is a zero column.
In fact, the computation may be stopped as soon as the upper matrix is in column echelon form: the remainder of the computation consists in changing the basis of the vector space generated by the columns whose upper part is zero.
For example, suppose that
![A=\left[ \begin{array}{cccccc}
1 & 0 & -3 & 0 & 2 & -8 \\
0 & 1 & 5 & 0 & -1 & 4 \\
0 & 0 & 0 & 1 & 7 & -9 \\
0 & 0 & 0 & 0 & 0 & 0 \end{array} \,\right].](http://upload.wikimedia.org/math/d/e/5/de56a2cf3e4a8a25494a76b8eaad5acd.png)
Then
![\left=
\left[\begin{array}{cccccc}
1 & 0 & -3 & 0 & 2 & -8 \\
0 & 1 & 5 & 0 & -1 & 4 \\
0 & 0 & 0 & 1 & 7 & -9 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\hline\\
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array}\right].](http://upload.wikimedia.org/math/2/b/8/2b86db5ea4854c7f5097fae0baca1e07.png)
Putting the upper part in column echelon form by column operations on the whole matrix gives
![\left=
\left[\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\hline\\
1 & 0 & 0 & 3 & -2 & 8 \\
0 & 1 & 0 & -5 & 1 & -4 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 & -7 & 9 \\
0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1
\end{array}\right].](http://upload.wikimedia.org/math/0/9/d/09d8904b3c5d6dc1776e6f3583c0284f.png)
The last three columns of B are zero columns. Therefore, the three last vectors of C,
are a basis of the null space of A.
Read more about this topic: Kernel (matrix)
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