Find the nullspace of the matrix 1 2 3 1 0 1 2 1 3 and write

Find the nullspace of the matrix (1 2 3 1 0 1 2 1 3) and write it as a span.

Solution

Let, x=[a b c]

Ax=0 gives

a+2b+3c=0

a+c=0 ie a=-c

2a+b+3c=0

Substituting a=-c in both these equations gives

-c+2b+3c=0

2b+2c=0 ie b=-c

-2c+b+3c=0 ie b=-c

a=-c, b=-c

x=(-c,-c,c)=c(-1,-1,1)

nullspace ={-1,-1,1}