Unitary Matrix mutually orthonormal vectors

[physics2000]

Homework Statement

Demonstrate that the columns of a unitary matrix form a set of mutually orthonormal vectors.

Homework Equations

hint - form the vectors $$u_i = {U_{ji}}$$ and $$u_k={U_{jk}}$$ from the $$i^{th}$$ and $$j^{th}$$ columns of $$U$$ and make use of the relationship $$U^{\dagger}U=I$$

The Attempt at a Solution

I thought my work was following the hint...but not sure...I know I need to end up with an identity matrix from the hint, in which it shows the 3 columns are (1,0,0) , (0,1,0), and (0,0,1), respectively...to show that I have a set of basis vectors in the unitary matrix...

[url=http://postimage.org/][PLAIN]http://s9.postimage.org/sw7ugplvz/photo_3.jpg[/url] upload[/PLAIN]

 

[Dick]

It looks like you've got it. The Kronecker delta symbol $\delta_{jk}$ is 1 if j=k and 0 if j≠k. Doesn't that give you your identity matrix?

 

[physics2000]

thanks,

I'm just a little confused because It looks like I only have one vector.

Do I need to turn that vector into a column, and for column 1 let i=k=1 , and for column 2 let i=k=3 and for column 3 let i=k=3, that would give:
[url=http://postimage.org/][PLAIN]http://s7.postimage.org/fj2bek7rf/photo_4.jpg[/url] free photo hosting[/PLAIN]

but I'm just not sure if my previous math boiled down to that, I feel like I'm missing something in the proof

 

[physics2000]

thanks,

would this be the next step? I feel like my notation is wrong or I did something wrong in the process

[url=http://postimage.org/][PLAIN]http://s16.postimage.org/4f3mski91/photo_5.jpg[/url] upload pics[/PLAIN]

 

[Dick]

thanks,

would this be the next step? I feel like my notation is wrong or I did something wrong in the process

Yes, it's wrong. You want to show that the inner products (your sums) are 1 or 0. That's doesn't show that the original matrix U is the identity. What does 'orthonormal' mean?