Question

Show that {(2,1,3),(1,2,3),(1,0,0)} is a basis of R^3.

Answer 1

put those vectors as columns in a matrix:

2 1 1
1 2 0
3 3 0

Then row reduce it.

If you get:

1 0 0
0 1 0
0 0 1

Then they are linearly independent and form a basis.