Proving the Pythagorean Theorem for Inner Product Spaces

In summary, the question asks to prove that for any vector v in an inner product space V, the norm of v is greater than or equal to the norm of the projection of v onto any finite dimensional subspace U of V. This can be shown using the Pythagorean Theorem and the properties of an inner product space, such as its basis vectors being orthonormal.
  • #1
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
 
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  • #2
playboy said:
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
[itex]\vec{V}[/itex] is a vector in an inner product space [itex]U[/itex]?
otherwise that doesn't make sense?
Maybe you can find some easy cases, and work from there?
 
  • #3
If you draw these two vectors out, it should be very obvious. The magnitude of the projection is just the component:

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

...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?
 
  • #4
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 [itex]w_i[/itex] of W. similarly write down the expansion of the norm in terms of the basis vectors [itex]u_i[/itex] of U.

What do you know about these two sets of basis vectors ?
 
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  • #5
okay thanks for the help everyone!

Gokul43201: ill try your metod and get give back my results!
 
  • #6
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?
 
  • #7
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" ?
playboy said:
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.

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

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

For the left hand side...
<u,u>^0.5
Am i on the right track?
Expand ||v|| similarly.
 
  • #8
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
 
  • #9
Gokul43201 said:
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 [itex]w_i[/itex] of W. similarly write down the expansion of the norm in terms of the basis vectors [itex]u_i[/itex] 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.
 
  • #10
playboy said:
Noooo... this is why its not making sense.
Isn't the projection...
<v wi>
______ wi
<wi wi>
and not
<v wi> wi
If [itex]w_i[/itex] is a basis vector, what's [itex]\langle w_i,w_i \rangle [/itex] ?
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.
 
Last edited:
  • #11
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
 

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors as input and produces a scalar value as output. It is used to measure the angle between two vectors and can also be used to calculate the length or magnitude of a vector.

2. How is an inner product calculated?

An inner product is calculated by taking the dot product of two vectors. This involves multiplying the corresponding components of the two vectors and then adding the results together. The resulting scalar value is the inner product of the two vectors.

3. What is the geometric interpretation of an inner product?

The geometric interpretation of an inner product is that it measures the projection of one vector onto another. This can be visualized as the shadow of one vector on the other when they are placed tail-to-tail.

4. What are some applications of inner products in science?

Inner products are widely used in physics and engineering to calculate work done, energy, and forces. They are also used in statistics and machine learning algorithms for data analysis and classification. In quantum mechanics, inner products are used to calculate probabilities and measure the overlap of quantum states.

5. How is an inner product different from a cross product?

An inner product is a scalar value, while a cross product is a vector. The inner product measures the angle between two vectors, while the cross product measures the area of the parallelogram formed by the two vectors. Additionally, the inner product is commutative, while the cross product is not.

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