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)
Famous quotes containing the word basis:
“Association with women is the basis of good manners.”
—Johann Wolfgang Von Goethe (1749–1832)
“That food has always been, and will continue to be, the basis for one of our greater snobbisms does not explain the fact that the attitude toward the food choice of others is becoming more and more heatedly exclusive until it may well turn into one of those forms of bigotry against which gallant little committees are constantly planning campaigns in the cause of justice and decency.”
—Cornelia Otis Skinner (1901–1979)
“Self-alienation is the source of all degradation as well as, on the contrary, the basis of all true elevation. The first step will be a look inward, an isolating contemplation of our self. Whoever remains standing here proceeds only halfway. The second step must be an active look outward, an autonomous, determined observation of the outer world.”
—Novalis [Friedrich Von Hardenberg] (1772–1801)