Question

Find a basis for the null space of the matrix.
A=[1 0 -5 0 -2; 0 1 5 0 2; 0 0 0 1 1; 0 0 0 0 0]

Answer 1

The basis for the null space of the matrix A=[1 0 -5 0 -2; 0 1 5 0 2; 0 0 0 1 1; 0 0 0 0 0] is null(A) = x2[-5 5 0 -1 0] + x4[-2 2 1 0 1].

Step-by-step explanation:

The basis of the null space of a matrix is determined by finding the solutions to the homogeneous system of equations given by setting the matrix equal to the zero vector. In this case, we can rewrite the given matrix A as:

[1 0 -5 0 -2;
0 1 5 0 2;
0 0 0 1 1;
0 0 0 0 0]

We can convert this matrix into row-echelon form to find the solutions. By performing row operations, we get:

[1 0 -5 0 -2;
0 1 5 0 2;
0 0 0 1 1;
0 0 0 0 0]

The row-echelon form shows that the second and fourth columns have leading ones. Therefore, the variables associated with those columns (x2 and x4) are the free variables. We can express the null space as:

null(A) = x2[-5 5 0 -1 0] + x4[-2 2 1 0 1]