Proving the Pythagorean Theorem for Inner Product Spaces

[playboy]

A question reads:

Let V be a vector in an inner product space V

show that ||v|| >= ||proj(u) v|| holds for all finitie dimensional subspaces of U.

Hint: Pythagorean Theorm.

Okay... where on Earth do i begin?

I thought perhaps I should expand the RIGHT side of the equation, but that dosn't seem to be getting me anywhere really.

The LEFT side seems pretty useless too, so I am stuck trying to show that the RIGHT side is >= ...

anybody have any ideas?

Thanks

 

[NateTG]

A question reads:
Let V be a vector in an inner product space V
show that ||v|| >= ||proj(u) v|| holds for all finitie dimensional subspaces of U.
Hint: Pythagorean Theorm.
Okay... where on Earth do i begin?
I thought perhaps I should expand the RIGHT side of the equation, but that dosn't seem to be getting me anywhere really.
The LEFT side seems pretty useless too, so I am stuck trying to show that the RIGHT side is >= ...
anybody have any ideas?
Thanks

Do you mean
\vec{V} is a vector in an inner product space U?
otherwise that doesn't make sense?
Maybe you can find some easy cases, and work from there?

 

[amcavoy]

If you draw these two vectors out, it should be very obvious. The magnitude of the projection is just the component:

\left|\text{proj}_{u}v\right|=\frac{\left(u,v\right)}{u}

...which is really just saying v*cosθ. What do you know about the cosine function? Well, for one, it is less than or equal to 1 for all values of θ. Does this help?

 

[Gokul43201]

apmcavoy : There are no "two vectors". A vector v in U is being projected onto some subspace (W, say) of U. However, with some additional construction, your method would work.

playboy : Call the subspace W, and write the projection down in terms of the basis vectors w_i of W. similarly write down the expansion of the norm in terms of the basis vectors u_i of U.

What do you know about these two sets of basis vectors ?

 

[playboy]

okay thanks for the help everyone!

Gokul43201: ill try your metod and get give back my results!

 

[playboy]

Now i am even more lost than ever :S

For the RHS...i get...

<v, w1> + <v, w2> + ...
||w1||...||w2||

For the left hand side...

<u,u>^0.5

Am i on the right track?

 

[Gokul43201]

First, we need to confirm that we're all solving the same problem. Where did U come from. Did you mean to write (in the OP) : "...for all finite dimensional subspaces U of the space V" ?

Now i am even more lost than ever :S
For the RHS...i get...
<v, w1> + <v, w2> + ...
||w1||...||w2||

The length (or norm) is not equal to the sum of its components.

proj_U(v) = \sum _{i=1}^k \langle v,w_i \rangle w_i

\implies ||proj_U(v)|| = \sqrt{\sum _{i=1}^k \langle v,w_i \rangle ^2}

For the left hand side...
<u,u>^0.5
Am i on the right track?

Expand ||v|| similarly.

 

[playboy]

Noooo... this is why its not making sense.

Isn't the projection...

<v wi>
______ wi
<wi wi>


and not


<v wi> wi



And on the LHS... (v is just a vecotr)
so ||v|| = (v1^2 + v2^2 + ... + vn^2)^0n5

 

[amcavoy]

apmcavoy : There are no "two vectors". A vector v in U is being projected onto some subspace (W, say) of U. However, with some additional construction, your method would work.

playboy : Call the subspace W, and write the projection down in terms of the basis vectors w_i of W. similarly write down the expansion of the norm in terms of the basis vectors u_i of U.

What do you know about these two sets of basis vectors ?

Yes, right. I'm sorry if that confused anyone who read this. I guess I was just assuming some vector u in U.

 

[Gokul43201]

Noooo... this is why its not making sense.
Isn't the projection...
<v wi>
______ wi
<wi wi>
and not
<v wi> wi

If w_i is a basis vector, what's \langle w_i,w_i \rangle ?

And on the LHS... (v is just a vecotr)
so ||v|| = (v1^2 + v2^2 + ... + vn^2)^0n5

What are v1, v2, etc ?

PS : Why are you not addressing the problem that no one really knows what the actual question is? The question as stated in the OP is incorrect and needs to be fixed.

 

[playboy]

The Question Reads:

Let v be a vector in an inner product space V

show that ||v|| >= ||proj u (v)|| holds for all finitie dimensional subspaces of U.

Hint: Pythagorean Theorm.
_____________________________________________
how i approached it:

Let (e1,...,en) be an orthonognal basis:

||proj u (V)||^2 = <proj u (V), proj u (V)>
||proj u (V)|| = ( <v e1>^2/<e1 e1> + ... + <v en>^2/<en en>)^0.5

Note that i did not show my full computation from the beginning, that would
just be too long to type out.

and ||v|| = (v1^2 + ... + Vn^2)^0.5 since v is a Vector

Im lost after this point :( Please help