Let A 6 6 6 6 6 6 6 6 Find a basis of nullspaceA SolutionA

Let A =|[|-6 6 6 -6 6 -6 -6 6]| Find a basis of nullspace(A)|. [|]|, [|]|, [|]|.

Solution

After applying row operations (R2= R1+R2) the given matrix can be wirtten as

[ -6 6 6 -6
   0 0 0 0 ]
Now write R1 as R1/(-6) we get

[ 1 -1 -1 1
0 0 0 0]
This is the reduced row echeleon form of the augmented matrix;

This is equivalnet to the equations :
x1 -x2 -x3 +x4 = 0
and 0x1 -0x2 -0x3 +0x4 = 0 which is simply 0=0

So x1= x2+x3-x4

This system has inifnite solutions as there are 4 vairables but only 1 equation to solve them
Thus x2, x3 and x4 take arbitrary values while x1 is = x2+x3-x4

Thus the soluion can be written in the vector form:

[ 1 1 0 0 ]^T c₁ + [ 1 0 1 0 ]^T c₂+ [ -1 0 0 1 ]^T c3

Hence enter 1, 1, 0, 0 in the first column of basis of null space(A)|

1, 0, 1, 0 in the second column
-1, 0, 0, 1 in the third column