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?