Let A 3 6 1 1 7 1 2 2 3 1 2 4 5 8 4 Find the basis for the

Let A = [-3 6 -1 1 -7 1 -2 2 3 -1 2 -4 5 8 -4]. Find the basis for the null space of A. Find the basis for the column space of A.

Solution

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:

c₂ +

c₄ +

c₅

Row Operation 1:

-3 6 -1 1 -7 1 -2 2 3 -1 2 -4 5 8 -4

multiply the 1st row by -1/3

1 -2 1  3 -1  3 7  3 1 -2 2 3 -1 2 -4 5 8 -4