Is there anyway to check I have an an orthogonal and/or orthonormal basis?

Question

I'm reading about Gram-Schmidt procedure in 3 dimensions. From what I understand the idea is to "fix" one of the vectors and alter the other 2 so they are all perpendicular. So say i have three vectors $u_1,u_2, u_3$ and I do Gram-Schimdt procedure to get $v_1,v_2,v_3$, firstly is there anyway to tell I have done the process correctly, and what is and how do I get a "orthonormal basis" from here, and once done can I check this step also? Thanks.

Answer 1

There is an easy test to see if your end result is valid. Arrange your vectors as column vectors of a matrix, and call that matrix $A$. Since you are looking for a basis, $A$ should be square (otherwise you have the wrong number of vectors). Now, compute $A^TA$. If you get a diagonal matrix (i.e., the only nonzero entries lie along the top-left to bottom-right diagonal), your vectors were orthogonal. If you get the identity matrix, your vectors were orthonormal.

To answer the other part of your question, once you have an orthogonal basis, you can turn it into an orthonormal basis by dividing each vector by its norm, i.e. find $\frac{v_i}{||v_i||}$ for each $i$.