How to easily prove that some vectors are a basis for the following:

Question

I know a matrix can have different basis but here I am not sure how to efficiently check which answer is correct. We have to be able to answer this in under 3 min so I can't check each one individualy? Do you have a trick to go faster than row reducing A?

Answer 1

So I followed @daniel's advice and I check if a corresponding element in A would be a lin combination and you realise pretty quickly that the 1st and last answer fail to be consistent.

Plus the 3rd answer does not make sense since col (A)^T is in R4 but the base is given in R3 so the only one left is Nul(A) which is easy to check by solving A(basisNUL) = 0 which works so we can confirm that the 2nd answer is correct!