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:
““The Love that dare not speak its name” in this century is such a great affection of an elder for a younger man as there was between David and Jonathan, such as Plato made the very basis of his philosophy, and such as you find in the sonnets of Michelangelo and Shakespeare. It is that deep, spiritual affection that is as pure as it is perfect.... It is in this century misunderstood ... and on account of it I am placed where I am now.”
—Oscar Wilde (1854–1900)
“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)
“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)