Kernel (matrix) - Basis

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].

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].

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].

The last three columns of B are zero columns. Therefore, the three last vectors of C,

\left,\; \left,\; \left

are a basis of the null space of A.

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