Using gram-schmidt to compute basis of column space.

Question

Say you want to find a basis for the column space of a matrix. Can you apply Gram-Schmidt to the columns of A and take the output as your basis? What is the most efficient method in general for finding the basis of a column space without using row reduction methods?

Answer 1

Gram-Schmidt will give you Orthonormal basis for the column space. This is more useful in most context . But Applying Gram-Schmidt to even a $4\times 4$ system is cumbersome enough. Even then you will need to start with a linearly independent set of column vectors.

If you just want to find a basis for column space then you can go by Replacement theorem.

That is you look at the first coulmn and the second one. If they are linearly dependent then you leave that and proceed to the third. If it is again linearly dependent then proceed to the fourth. If you exhaust everything you have rank $1$ and the first coulmn is itself enough to span .

If the second one was linearly independent. Then you look at the linear independence of the first second and third column together. and then proceed similarly.

However the Row Reduction Method(Applied to the transpose) is probably the most efficient to find the basis for the column space.

Answer 2

The problem with Gram-Schmidt here is that it is intended to take a linearly independent set and turn it into an orthonormal set. This would mean that here you're supposed to already start with a basis, and then simply turn it into an ON-basis. However it could still work, as once you have a basis, the possibly remaining vectors will just become zero in the process. So while it might possibly work, it is not really what it is intended for, and row reduction is probably the best way to go.