Inner Product Space - Pythagorean?

  • #1
ElijahRockers
Gold Member
270
10

Homework Statement


Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##:

||v - vo||^2 = ||v||^2 - ||vo||^2

Homework Equations


General inner product space properties, I believe.

The Attempt at a Solution


So Vo is finite dimensional and has an orthonormal basis {v1, v2,...., vn} for finite n. I can say that right? If so,

##v_0 = \sum_{i=1}^{n}<v,v_i>v_i## (Not sure if that helps really)

Also <v - vo , vo> = 0 since they are orthogonal.

That's about where I get stuck. I see that the expression I'm supposed to arrive at is basically pythagorean theorem for inner product spaces, and I can see it in my head quite easily if n is 2 dimensional.

So I tried working backwards:

||v - vo||^2 = ||v||^2 - ||vo||^2

||v - vo||^2 + ||vo||^2 = ||v||^2

And using inner product properties I can get:

<v - vo , v - vo> + <vo, vo> = <v , v>

I don't know if that really helps though.

Any advice is appreciated.
 

Answers and Replies

  • #2
pasmith
Homework Helper
1,961
595

Homework Statement


Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##:

||v - vo||^2 = ||v||^2 - ||vo||^2

Homework Equations


General inner product space properties, I believe.

The Attempt at a Solution


So Vo is finite dimensional and has an orthonormal basis {v1, v2,...., vn} for finite n. I can say that right? If so,

##v_0 = \sum_{i=1}^{n}<v,v_i>v_i## (Not sure if that helps really)

Also <v - vo , vo> = 0 since they are orthogonal.

That's about where I get stuck. I see that the expression I'm supposed to arrive at is basically pythagorean theorem for inner product spaces, and I can see it in my head quite easily if n is 2 dimensional.

So I tried working backwards:

||v - vo||^2 = ||v||^2 - ||vo||^2

||v - vo||^2 + ||vo||^2 = ||v||^2

And using inner product properties I can get:

<v - vo , v - vo> + <vo, vo> = <v , v>

I don't know if that really helps though.
Expand [itex]\|v - v_0\|^2 = \langle v - v_0, v - v_0 \rangle[/itex]. You should get four terms. Why do the two terms you don't want vanish?
 
  • #3
ElijahRockers
Gold Member
270
10
Didn't think I could...

So would it be:
##<v,v> -2<v,v_0 > + 2<v_0 , v_0> = <v, v>##
##<v,v_0 > = <v_0, v_0>##
##v = v_0##
##v - v_0 = 0##

so ##<v - v_0 , v_0> = 0##

Is that right?
 
  • #4
Dick
Science Advisor
Homework Helper
26,260
619

Homework Statement


Let ##V## be an inner product space and let ##V_0## be a finite dimensional subspace of ##V##. Show that if ##v ∈ V## has ##v_0 = proj_{V_0}(v)##:

||v - vo||^2 = ||v||^2 - ||vo||^2

Homework Equations


General inner product space properties, I believe.

The Attempt at a Solution


So Vo is finite dimensional and has an orthonormal basis {v1, v2,...., vn} for finite n. I can say that right? If so,

##v_0 = \sum_{i=1}^{n}<v,v_i>v_i## (Not sure if that helps really)

Also <v - vo , vo> = 0 since they are orthogonal.

That's about where I get stuck. I see that the expression I'm supposed to arrive at is basically pythagorean theorem for inner product spaces, and I can see it in my head quite easily if n is 2 dimensional.

So I tried working backwards:

||v - vo||^2 = ||v||^2 - ||vo||^2

||v - vo||^2 + ||vo||^2 = ||v||^2

And using inner product properties I can get:

<v - vo , v - vo> + <vo, vo> = <v , v>

I don't know if that really helps though.

Any advice is appreciated.
You say you can show <v - vo , v - vo> + <vo, vo> = <v , v>? Isn't that the same thing as ||v - vo||^2 + ||vo||^2 = ||v||^2?
 
  • #5
ElijahRockers
Gold Member
270
10
You say you can show <v - vo , v - vo> + <vo, vo> = <v , v>? Isn't that the same thing as ||v - vo||^2 + ||vo||^2 = ||v||^2?
Yea. but as far as I am able to tell I'm supposed to be able to show that given vo = proj_Vo (v). I was working backwards from ||v - vo||^2 etc
 
  • #6
Dick
Science Advisor
Homework Helper
26,260
619
Yea. but as far as I am able to tell I'm supposed to be able to show that given vo = proj_Vo (v). I was working backwards from ||v - vo||^2 etc
How did you define projection? Have you shown <v-vo,vo>=0? That's all you really need. Expand ||v|| = ||(v-vo)+vo||.
 
  • #7
ElijahRockers
Gold Member
270
10
How did you define projection? Have you shown <v-vo,vo>=0? That's all you really need. Expand ||v|| = ||(v-vo)+vo||.
Yes, I defined that in my original post. That's what I was thinking! Thanks!

EDIT: Shown, not defined.
 

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