# Thread: Use Gram-Schmidt process to find an orthogonal basis for the image space A. QR Factor

1. ## Use Gram-Schmidt process to find an orthogonal basis for the image space A. QR Factor

Question is attached. I just would like someone to verify whether the answer involves fractions/mixed numbers because I do tend to get antsy when I see them in my work. I wonder if I did it right.

So anyway here is what I got with Gram-Schmidt Process:

With x1, x2, and x3 representing the first, second, and third columns of A and v1, v2, and v3 representing each vector in the orthogonal basis respectively,

v1= [1 2 0 2] v2 = [x2] and v3= [-1 7/6 1/6 -2/3]

Correct? And since we are on the topic, how do I go about finding the QR factorization of A ?

2. ## Re: Use Gram-Schmidt process to find an orthogonal basis for the image space A. QR Fa

I can't see how you can hope to get unit length vectors b]without[/b] fractions. In fact, it is only because these numbers are chosen carefully that you don't get irrational numbers!

3. ## Re: Use Gram-Schmidt process to find an orthogonal basis for the image space A. QR Fa

Hello,

This type of problem is part of a set of problems that try to avoid fractions. I know with gram-schmidt problems fractions tend to be inevitable but I was just wondering if I had done the problem correctly.

4. ## Re: Use Gram-Schmidt process to find an orthogonal basis for the image space A. QR Fa

It's not asking for orthonormal, so you can construct an orthogonal basis that only involves whole numbers. I got the same v1 and v2 as you did but after scaling to whole number entries, I got [26 31 -75 -80]T for v3