Let A Find a basis of KerA 0 9 6 3 0 6 4 2 SolutionThis yiel

Let A=

Find a basis of Ker(A).

Solution

This yields the equation

The matrix equation above is equivalent to the following homogeneous system of equations

We now transform the coefficient matrix of the homogeneous system above to the reduced row echelon form to determine the solution space.

can be transformed by a sequence of elementary row operations to the matrix

The reduced row echelon form of the augmented matrix is

which corresponds to the system

The leading entries in the matrix have been highlighted in yellow.

A leading entry on the (i,j) position indicates that the j-th unknown will be determined using the i-th equation.

Those columns in the coefficient part of the matrix that do not contain leading entries, correspond to unknowns that will be arbitrary.

The system has infinitely many solutions:

The solution can be written in the vector form:

Therefore the kernel of L has a basis formed by the set