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:

    “Far from being the basis of the good society, the family, with its narrow privacy and tawdry secrets, is the source of all our discontents.”
    —Sir Edmund Leach (20th century)

    “The cultivation of literary pursuits forms the basis of all sciences, and in their perfection consist the reputation and prosperity of kingdoms.”
    —Marquês De Pombal (1699–1782)

    “The basis of our governments being the opinion of the people, the very first object should be to keep that right; and were it left to me to decide whether we should have a government without newspapers, or newspapers without a government, I should not hesitate a moment to prefer the latter.”
    —Thomas Jefferson (1743–1826)