2 Let A a Find a basis for the null space of A b Find the ra

2. Let A= (a) Find a basis for the null space of A. (b) Find the rank and the nullity of A.

Solution

Step 1: Transform the matrix to the reduced row echelon form  (Hide details)

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:

c₃

The rank of matrix will be 3 since there are 3 leading ones and nullity can be found using Ax = 0 which gives nullity as 2

Row Operation 1:

1 2 3 4 0 1 1 2 1 3 4 7 0 2 2 6

add -1 times the 1st row to the 3rd row

1 2 3 4 0 1 1 2 0 1 1 3 0 2 2 6