Required to find an orthonormal basis for the following subspace of R4
I know that to find the othonormal basis, it is required that i find the basis for the subspace, then I use Gram Schmidt process. Afterwards Ill normalize the vectors I get from the GS process and that should give me the orthonormal basis. My question really is, can I get more than one basis for the subspace? I got $\{(3, -2, 1, 0) , ( 0, 1, 0, 1)\}$ whereas my teacher got $\{(-1, -2, 1, 0), (0, 1, 0, 1) \}$. Note: we both used different free variables.
Also, If i used x and y as free variables and get my basis, then i use x and z as free variables to get my basis. They both are legit bases right ?
Best Answer
Think of $\mathbb R^2$ - the real plane - since your subspace clearly has dimension $2$ and is therefore a plane.
Take an arbitrary unit vector at the origin. Can you find a second unit vector at right-angles to the first?
That is essentially the configuration you are looking for. It doesn't compute the basis vectors for you, but I hope it helps you to visualise the problem.