Inner products and orthogonal basis

In summary, the conversation is about finding and proving an orthogonal basis relative to an inner product. The suggestion is to use the Gram-Schmidt Algorithm to construct an orthogonal basis, and to check if the basis is orthogonal by using the inner product to confirm that the vectors are at right angles. It is also noted that changing the inner product can change the metric of the vectors.
  • #1
NullSpaceMan
8
0
Hi all!

This looks a pretty nice forum. So here's my question:

How do I find/show a basis or orthogonal basis relative to an inner product? The reason I ask, is because in my mind I see the inner product as a scalar, and thus I find it difficult to "imagine" how a scalar lives in a space.

Many thanks! I would like to discuss.

Have a good one:cool:
 
Physics news on Phys.org
  • #2
Try the The Gram-Schmidt Algorithm - it will construct an basis orthogonal wrt your inner product given any other basis.

The inner product of two vectors is a scalar (it is in a sense the angle between the two vectors) - what exactly are you trying to imagine?
 
  • #3
What river_rat said. If you have a set of linearly independent vectors that span the inner product space, you can use the Gram-Schmidt orthogonalisation process to find an orthonormal basis of identical span.

The inner-product is nothing more than a bilinear form defined over any vector space. It helps to distinguish the operation from the vector space itself, since all "inner-product spaces" are vector spaces without their respective inner products as well.EDIT: Typo correction
 
  • #4
Thanks.

I am familiar with the grahm-schmidt algorithm, but I was wondering how I would go about proving a given basis is orthogonal relative to an inner product? How do I picture such in my head?

thanks again,

:cool:
 
  • #5
Well check if [tex] \vec{e_i} \cdot \vec{e_j} = 0 [/tex] [tex] \forall i \neq j [/tex]. If that is true then your basis is orthogonal relative to that innerproduct. For R^2 a non standard inner product amounts to declaring some other angle to mean " at [tex] \frac{\pi}{2} [/tex] " - so all you have done is shift your axis so that they are no longer meet at right angles (relative to the normal inner product that is, changing the inner product also changes the metric - so you are squishing some directions and expanding others)
 
Last edited:

1. What is an inner product?

An inner product is a mathematical operation that takes two vectors as input and produces a scalar (number) as output. It measures the similarity or "closeness" between two vectors in a vector space.

2. What is the significance of an inner product?

Inner products have many applications in mathematics, physics, and engineering. They are used to define and measure various quantities such as length, angle, and distance between vectors. They also play a crucial role in defining orthogonal bases and determining the orthogonality of vectors.

3. What is an orthogonal basis?

An orthogonal basis is a set of vectors that are mutually perpendicular (orthogonal) to each other and have a length of 1 (unit vectors). This means that the inner product between any two vectors in an orthogonal basis is zero, and the vectors span the entire vector space.

4. How do you find an orthogonal basis?

To find an orthogonal basis, one can use the Gram-Schmidt process, which involves taking a set of linearly independent vectors and transforming them into an orthogonal set by subtracting their projections onto each other. Alternatively, one can use techniques such as the QR decomposition or the Singular Value Decomposition (SVD) to find an orthogonal basis.

5. What are some real-life applications of inner products and orthogonal bases?

Inner products and orthogonal bases have many practical applications. For example, in image and signal processing, they are used for data compression and noise reduction. In quantum mechanics, they are used to describe the properties and behavior of particles. In engineering, they are used in fields such as control systems, signal processing, and machine learning.

Similar threads

  • Linear and Abstract Algebra
Replies
9
Views
367
Replies
2
Views
935
Replies
3
Views
2K
  • Linear and Abstract Algebra
2
Replies
43
Views
5K
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
276
  • Math POTW for University Students
Replies
2
Views
830
  • Linear and Abstract Algebra
Replies
3
Views
177
  • Linear and Abstract Algebra
Replies
2
Views
917
  • Quantum Physics
Replies
8
Views
2K
Back
Top