[Math] Orthogonal basis and projection

Question

I was hoping someone could explain to me using the vectors below and $V=Span\{v_1,v_2,v_3\}$ how I could determine the orthogonal basis for $V$ and the projection of $y$ onto $V$. I have been googling for examples / explanations but I'm honestly completely stuck.

Thank you very much 🙂

Answer 1

Since $dim(V)=3$, you already have a basis of $V$ : $v_1,v_2,v_3$.

You just need to make it orthogonal, using Gram–Schmidt process : $$u_1=v_1$$ $$u_2=v_2-(v_2,u_1)u_1$$ $$u_3=v_3-(v_3,u_2)u_2-(v_3,u_1)u_1$$

Then : $$proj_V(y) = \frac{(y,u_1)}{||u_1||^2}u_1+\frac{(y,u_2)}{||u_2||^2}u_2+\frac{(y,u_3)}{||u_3||^2}u_3$$