I have the following problem along with my solution.

My textbook lists the correct answer as

Is my solution correct? So, once I arrive at the reduced-row echelon form of the matrix A, I can tell which vectors in the set are linearly independent by the "leading ones" in my matrix A. Those vectors will always form a basis?

I have two vectors in R^4 and my professor's answer was two vectors in R^4. The book lists two vectors in R^2. Is it possible to form a basis for the subspace that has some combination of vectors in R, R^3, R^5, R^n?

My professor transposes the matrix A and then proceeds to find a basis. He arrives at a different basis. There are infinitely many basis correct? My professor said that he transposes A so that he can use row operations and not change the solution. Well, if there are infinitely many basis, why does this matter? Could someone please clarify this problem for me?

Thanks