Find a basis for a subspace

[Fanta]

say we are given a subspace like this:

Being W the subspace of R generated by (1,-2,3,-1), (1,1,-2,3) determine a basis and the dimension of the subspace.

Won't the vectors given work as a basis, as long as they are linearly independent?
If so, all we have to do is check for dependance, and if the system is dependent, we would choose one of the vectors as a base, and the dimention would be 1. if it was independent, the basis would be both of those vectors, and the dimention would be 2.

By the same logic, If I was asked to determine the dimention of the subspace generated by those vectors, i'd just check for linear dependance.
Is this right?

By the way, should I simplify the basis by putting them in a matrix and getting it to row-echelon form? would the basis still "work" after that?

 

[Mark44]

The two vectors are obviously linearly independent; if they were dependent, one would be a constant multiple of the other. If you had been given three vectors, you wouldn't be able to tell by inspection that the three vectors were linearly independent.

Since the two vectors give generate the subspace in question, they span the subspace, and hence form a basis for it.

The dimension of a vector subspace is equal to the number of vectors in the spanning set for the subspace.