- #1

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## 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.