Question

Given
f(e1)=(1,−1,0),f(e2)=(0,1,−1),f(e3)=(1,0,−1)
. What is the rank and nullity?

Answer 1

you want to create a matrix using those vectors as columns:

1 0 1
-1 1 0
0 -1 -1

Then row reduce it. The number of pivots present is the rank, and the number of free variables is the nullity (i think, havent seen that terminology before)

Answer 2

when you row reduce it you get:

1 0 1
0 1 1
0 0 0

There are two pivots, and one free variable. so the rank is 2, nullity is 1.