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:
“Knighterrantry is a most chuckleheaded trade, and it is tedious hard work, too, but I begin to see that there is money in it, after all, if you have luck. Not that I would ever engage in it, as a business, for I wouldn’t. No sound and legitimate business can be established on a basis of speculation. A successful whirl in the knighterrantry line—now what is it when you blow away the nonsense and come down to the cold facts? It’s just a corner in pork, that’s all.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“Our fathers and grandfathers who poured over the Midwest were self-reliant, rugged, God-fearing people of indomitable courage.... They asked only for freedom of opportunity and equal chance. In these conceptions lies the real basis of American democracy. They and their fathers give a genius to American institutions that distinguished our people from any other in the world.”
—Herbert Hoover (1874–1964)
“It was the custom
For his rage against chaos
To abate on the way to church,
In regulations of his spirit.
How good life is, on the basis of propriety,
To be followed by a platter of capon!”
—Wallace Stevens (1879–1955)