You have 3 vectors from `RR^3` if all 3 vectors are linearly independent then they make a base for `RR^3` that is they span `RR^3`.
Vectors `v_1,v_2,ldots,v_n` are linearly independent if
`a_1 v_1+a_2 v_2+cdots+a_n v_n=0`
Let's check if our vectors are linearly independent.
From this we get system of 3 equations (one for each coordinate) with 3 variables.
Now you can solve this system of equations by using Gauss elimination.
System has unique solution `a=b=c=0` which means that all three vectors are independent and thus they span `RR^3` ` ` which is what we were supposed to prove.