This is a follow up question ( math.stackexchange.com/q/3018473); i'm interested in understanding some other part of the problem.
I have three vectors, v1, v2, v4, which are linearly independent.
I need to find the orthonormal basis of their linear span.
To see if v1 and v2 are orthonormal, I do the dot multiplication using u1 and v2:
(u1,v2)= 0 , as shown in the picture.
Now here is were my question arises. I found u3, which gives me u3=(0,0,1,0).
Since before , to see if u1 and v2 are orthogonal, i found the dot product of (u1,v2), can i do the same thing with v3?
Meaning:
(u3·v1)=0 and (u3·v2)=0? (i check the answers using the same multiplication process in the images)
Practically my question is: since i found the dot product of (u1,v2) to see that they are orthogonal to each other , can i do the dot product of (u3,v1) and (u3,v2) to see if the u3 is orthogonal to v1 and v2?
(u3,v1)= (0,0,1,0)·(1,1,0,0)=0
(u3,v2)=(0,0,1,0)·(1,-1,0,0)=0
Somebody please help?
Best Answer
The answer is yes, if you have two non-zero vectors ${\bf a}$ and ${\bf b}$ and want to decide whether they are orthogonal, you just need to calculate their inner product (dot product). If it is zero then they are orthogonal
$$ \langle {\bf a} | {\bf b} \rangle = 0 ~~~{\bf a}\mbox{ and }{\bf b}\mbox{ are orthogonal} $$
So in your example, $\langle {\bf u}_1 | {\bf v}_2 \rangle = 0$ which means that ${\bf u}_1$ and ${\bf v}_2$ are orthogonal. Now, since $\langle {\bf u}_3 | {\bf v}_1 \rangle = 0$ then ${\bf u}_3$ is also orthogonal to ${\bf v}_1$