Inner Product

  • Thread starter phagist_
  • Start date
  • #1
25
0

Homework Statement


Let v,w be vectors in a complex inner product space such that ||v|| = 1,
||w|| = 3 and <v,w> = 1 + 2i. Find ||v + iw||.


Homework Equations


The properties of an inner product.


The Attempt at a Solution


I figured
[tex]||v+iw||^2[/tex] = <v+iw,v+iw>

Then using the properties of the inner product, I broke it up;

[tex]||v+iw||^2[/tex] = <v,v+iw>+<iw,v+iw>

and <v,v+iw> = <v,v> + <v,iw> and using the 'conjugate symmetry'
= <v,v> - i<v,w>
= 1-i(1+2i)
= 3-i

Now for <iw,v+iw>;
=i<w,v+iw>
=i{<w,v>+<w,iw>}
=i{1-2i - i<w,w>} (since <v,w>=[tex]\overline{<w,v>}[/tex])
=i+2-9i
=2-8i

Now combining it all i get [tex]||v+iw||^2[/tex] = 5-9i

But this is supposed to be the square of the magnitude of v+iw.. so it should be a real number right?

Can somebody help point out where I have gone wrong?
Cheers.

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
41,847
964

Homework Statement


Let v,w be vectors in a complex inner product space such that ||v|| = 1,
||w|| = 3 and <v,w> = 1 + 2i. Find ||v + iw||.


Homework Equations


The properties of an inner product.


The Attempt at a Solution


I figured
[tex]||v+iw||^2[/tex] = <v+iw,v+iw>

Then using the properties of the inner product, I broke it up;

[tex]||v+iw||^2[/tex] = <v,v+iw>+<iw,v+iw>

and <v,v+iw> = <v,v> + <v,iw> and using the 'conjugate symmetry'
= <v,v> - i<v,w>
= 1-i(1+2i)
= 3-i

Now for <iw,v+iw>;
=i<w,v+iw>
=i{<w,v>+<w,iw>}
=i{1-2i - i<w,w>} (since <v,w>=[tex]\overline{<w,v>}[/tex])
=i+2-9i
Here is your error. i(1- 2i- 9i)= i+ 2+ 9. You forgot the second i on the last term.

=2-8i

Now combining it all i get [tex]||v+iw||^2[/tex] = 5-9i

But this is supposed to be the square of the magnitude of v+iw.. so it should be a real number right?

Can somebody help point out where I have gone wrong?
Cheers.

Homework Statement





Homework Equations





The Attempt at a Solution

 
  • #3
25
0
oops, thanks a lot halls!
 

Related Threads on Inner Product

  • Last Post
Replies
6
Views
1K
  • Last Post
Replies
13
Views
1K
  • Last Post
Replies
5
Views
1K
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
8
Views
1K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
3
Views
17K
  • Last Post
2
Replies
25
Views
3K
P
Top