Question about linear mappings and inner product spaces

In summary, the problem at hand is to prove the existence of a linear mapping on an inner product space with two inner products, such that the mapping satisfies a specific property. The key to solving this problem is to use an orthonormal basis and reformulate the problem in terms of matrices. A simple solution involves using two different orthonormal bases, one for each inner product.
  • #1
34567
1
0

Homework Statement



Hi, I am having difficulty with the following proof:

Let V be an inner product space (real of dimension n) with two inner products in V, <,> and [,]. Prove that there exists a linear mapping on V such that [L(x),L(y)] = <x,y> for all x,y in V.

I am stuck as to where to go with the proof. I know that I need to construct a linear mapping with the above property, however I'm not sure where to go from there. Any insight into this would be appreciated.
Thanks


Homework Equations





The Attempt at a Solution


 
Physics news on Phys.org
  • #2
Hint: let ##\{e_1,\ldots,e_n\}## be an orthonormal (with respect to ##\langle , \rangle##) basis for ##V##. It suffices to find a linear map ##L## such that
$$[L(e_i), L(e_j)] = \langle e_i, e_j\rangle = \delta_{ij}$$
One way to proceed from here is to reformulate the problem in terms of matrices.
 
  • #3
There's a simple solution that involves two orthonormal bases, one that's orthonormal with respect to <,>, and one that's orthonormal with respect to [,].
 

1. What is a linear mapping?

A linear mapping, also known as a linear transformation, is a function that preserves the operations of vector addition and scalar multiplication. This means that when you apply a linear mapping to a vector, the resulting vector will also satisfy the properties of vector addition and scalar multiplication.

2. What is an inner product space?

An inner product space is a vector space equipped with an inner product, which is a mathematical operation that takes in two vectors and outputs a scalar value. This inner product satisfies certain properties such as symmetry, linearity, and positive-definiteness.

3. How do you determine if a mapping is linear?

To determine if a mapping is linear, you can use the linearity property, which states that for any two vectors u and v and any scalar c, the mapping of the sum of u and v is equal to the sum of the mappings of u and v, and the mapping of c times u is equal to c times the mapping of u. If these properties hold true, then the mapping is linear.

4. What is the difference between a linear mapping and a linear transformation?

The terms linear mapping and linear transformation are often used interchangeably, but some mathematicians make a distinction between the two. Linear mapping refers to the function itself, while linear transformation refers to the transformation of the underlying vector space.

5. How are inner product spaces used in real life applications?

Inner product spaces have many applications in fields such as physics, computer science, and engineering. For example, in physics, inner product spaces are used to study quantum mechanics and the behavior of particles. In computer science, inner product spaces are used in data analysis and machine learning algorithms. In engineering, inner product spaces are used in signal processing and control systems.

Similar threads

  • Calculus and Beyond Homework Help
Replies
0
Views
419
  • Calculus and Beyond Homework Help
Replies
14
Views
532
  • Calculus and Beyond Homework Help
Replies
24
Views
673
  • Calculus and Beyond Homework Help
Replies
5
Views
844
  • Calculus
Replies
4
Views
379
  • Calculus and Beyond Homework Help
Replies
3
Views
109
  • Calculus and Beyond Homework Help
Replies
5
Views
3K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
Back
Top