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.

Read more about this topic:  Kernel (matrix)

Famous quotes containing the word basis:

    “The primacy of the word, basis of the human psyche, that has in our age been used for mind-bending persuasion and brain-washing pulp, disgraced by Goebbels and debased by advertising copy, remains a force for freedom that flies out between all bars.”
    —Nadine Gordimer (b. 1923)

    “The basis of art is truth, both in matter and in mode.”
    —Flannery O’Connor (1925–1964)

    “All costumes are caricatures. The basis of Art is not the Fancy Ball.”
    —Oscar Wilde (1854–1900)