# Orthogonal Complements and conjugate of inner product

• Dec 1st 2011, 05:55 PM
maxgunn555
Orthogonal Complements and conjugate of inner product
Let V be R
3
with the inner product
<v, w> = 3v1w1 + 2v2w2 + v3w3
where v = (v1, v2, v3) and w = (w1, w2, w3).
Find the orthogonal complement of the subspace U where
U = {(v1, v2, v3) element of V such that 2v1 - v2 = v3}.

Hi i understand that an orthogonal complement space it looks like this:

U complement = (v element of V : <u, v> = 0 for all u element U)

and that:
<v+w,u>=<v,u>+<w,u>=0+0=0

they just havent given me any usefull examples and only 2 examples. so help is needed please thnx
• Dec 1st 2011, 05:57 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
and forgot to ask what does this 'conjugate' of a vector refer to ie: for w = (w1, w2) what would 1w¯ ie the conjugate of w1 look like? also then how would conjugating the inner product space of a w = (w1, w2) and v = (v1,v2) change it?

this will help me answer my question on hermitian inner product ty.
• Dec 1st 2011, 06:17 PM
Deveno
Re: Orthogonal Complements and conjugate of inner product
for a complex vector space, if $\displaystyle v = (v_1,v_2,\dots,v_n), \overline{v} = (\overline{v_1},\overline{v_2},\dots,\overline{v_n })$, where $\displaystyle \overline{v_j}$ is the complex-conjugate of the complex number $\displaystyle v_j$. note that for a real vector space, this doesn't do anything, since real numbers are their own conjugates.

i would help further with your first question, but the notation is hopelessly confusing to me. perhaps you should read the forum guide on how to use LaTex here:
http://www.mathhelpforum.com/math-he...orial-266.html

(note: we now use [tex] tags, not [tex] tags, but the usage inside the tags is still the same).
• Dec 1st 2011, 06:21 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
thnx for the quick answer. yeah ive edited it somewhat now.
• Dec 1st 2011, 06:23 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
also i get it's the complex conjugate but basically how do i go about proving that <v, w> = 2v1w¯1 + 2v1w¯2 + 2v2w¯1 + 4v2w¯2 is a hermitian inner product. ie the complex conjugate of the inner product i need to find to find symmetry. as well as definiteness and positivity.
• Dec 1st 2011, 06:31 PM
Deveno
Re: Orthogonal Complements and conjugate of inner product
well, you need to show 3 things:

1. conjugate symmetry, that is: $\displaystyle \langle v,w \rangle = \overline{\langle w,v \rangle}$
2. linearity in the first argument: $\displaystyle \langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle; \langle \alpha v,w \rangle = \alpha \langle v,w \rangle$
3. positive-definiteness: $\displaystyle \langle v,v \rangle \geq 0; \langle v,v \rangle = 0 \implies v = 0$

it may be helpful to know that $\displaystyle v_j\overline{v_j} = |v_j|^2$ which is always a positive real number.
• Dec 1st 2011, 06:36 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
i know i was given those 3 properties but no examples...
• Dec 1st 2011, 06:42 PM
Deveno
Re: Orthogonal Complements and conjugate of inner product
well, let's look at the (supposed) inner product you're given:

$\displaystyle \langle v,w \rangle = 2v_1\overline{w_1} + 2v_1\overline{w_2} + 2v_2\overline{w_1} + 4v_2\overline{w_2}$.

what is $\displaystyle \overline{\langle v,w \rangle}$? (hint: use the fact that $\displaystyle \overline{a+b} = \overline{a} + \overline{b}$ and $\displaystyle \overline{\overline{a}} = a$).
• Dec 1st 2011, 08:10 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
oh right the conjugate of inner product jsut switches which vector w or v has the conjugate over it. i still dont know what you have to do with a conjugate if they dont actually give you any complex numbers in there.
• Dec 1st 2011, 08:11 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
i should probably come back to this when im thinking better. working through the night isnt doing me any favours. i take it yuo're in america then. thnx.
• Dec 4th 2011, 11:02 AM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
so is it just to do with switching the conujgates ie w1 conjugate becomes w1 whilst v1 becomes v1 conjugate. if so that doesnt look like it would be equal to the regular inner product...
• Dec 4th 2011, 11:24 AM
Deveno
Re: Orthogonal Complements and conjugate of inner product
it's NOT the same as the regular "dot product". but for REAL vector spaces, a real number IS it's own conjugate, so then we DO get a "standard inner product". see, we want to be able to squeeze a real number out of the inner product of two complex vectors (or else "positive definite" won't even make any sense), and that pretty much forces us to use conjugates.

so, yes, when you switch the order of v and w, you wind up with the conjugate of the original inner product.
• Dec 4th 2011, 12:20 PM
maxgunn555
Re: Orthogonal Complements and conjugate of inner product
ok kl i still dont know how to show the winding up of the inner product without actually putting in some complex numbers. but regardless i^2 does squeeze out a real number. i dont know how to show the definite though i was able to show the positivity. or is it that definiteness is just a given once you've proven positivity? thnx
• Dec 4th 2011, 05:20 PM
Deveno
Re: Orthogonal Complements and conjugate of inner product
for a complex number z = a+ib, $\displaystyle z\overline{z} = (a + ib)(a - ib) = a^2 - (-b^2) + i(ab - ab) = a^2 + b^2$, which is not only real, but non-negative, and only 0 for z = 0 = 0+i0.