Math Help - find a vector in an orthnormal basis

1. find a vector in an orthnormal basis

suppose I have 3 vectors, a b and c that form an orthonormal basis. How would I find a vector v = (1,2,3) in the a b c system? I don't really understand what its asking. Do I have to find a linear combination from a b c that forms v?
so basically for some scalars x y and z, I have to find

xa + by + cz = v?

if I'm right, how would I go about doing that?

2. Re: find a vector in an orthnormal basis

Originally Posted by Kuma
suppose I have 3 vectors, a b and c that form an orthonormal basis. How would I find a vector v = (1,2,3) in the a b c system? I don't really understand what its asking. Do I have to find a linear combination from a b c that forms v?
so basically for some scalars x y and z, I have to find

xa + by + cz = v?

if I'm right, how would I go about doing that?
That's the beauty of the orthonormal bases. We know that $v=xa+by+cz$ for some $x,y,z\in\mathbb{C}$ but then $\langle v,a\rangle=x\langle a,a\rangle+y\langle b,a\rangle+z\langle c,a\rangle=x\cdot1 +y\cdot0+z\cdot 0$ and so $\langle v,a\rangle=x$. Doing the same analysis we arrive at the fact that $v=\langle v,a\rangle a+\langle v,b\rangle b+\langle v,c\rangle c$.

3. Re: find a vector in an orthnormal basis

wait I'm a little lost. How did x(a,a) turn into x? (a,a) is just ((a1,a2,a3),(a1,a2,a3)) in R3. So its just a set of vectors. How did it turn into 1? Unless I'm misunderstanding. What do the pointy brackets mean?

4. Re: find a vector in an orthnormal basis

Originally Posted by Kuma
wait I'm a little lost. How did x(a,a) turn into x? (a,a) is just ((a1,a2,a3),(a1,a2,a3)) in R3. So its just a set of vectors. How did it turn into 1? Unless I'm misunderstanding. What do the pointy brackets mean?
The angle brackets mean inner product, i.e. $\langle a,a\rangle=a\cdot a=1$. Make sense now?

5. Re: find a vector in an orthnormal basis

ahh ok. Thanks. Although I think I got the answer through row reduction before your post. Basically I made a matix with the vectors abc and set them equal to v. I got a unique scalar solution for x y and z Those scalars multiplied by a b and c gives me v.

6. Re: find a vector in an orthnormal basis

Originally Posted by Kuma
ahh ok. Thanks. Although I think I got the answer through row reduction before your post. Basically I made a matix with the vectors abc and set them equal to v. I got a unique scalar solution for x y and z Those scalars multiplied by a b and c gives me v.
That works too haha