Inner product space - minimization.

In summary, the conversation discusses finding the linear polynomial g(t) that is closest to the function f(t) = e^t in the vector space C[1,1] of continuous real valued functions on the interval [1,1]. The solution involves finding orthogonal basis vectors u1 and u2 for a subspace S in C[-1,1] and using them to calculate the affine function g(t) that minimizes the integral for f(t). The basis vectors used in the solution are 1 and t.
  • #1
binbagsss
1,259
11
The question is : If the vector space C[1,1] of continuous real valued functions on the interval [1,1] is equipped with the inner product defined by (f,g)=[itex]^{1}_{-1}[/itex] [itex]\intf(x)g(x)dx[/itex]

Find the linear polynomial g(t) nearest to f(t) = e^t?

So I understand the solution will be given by (u1,e^t).||u1|| + (u2,e^t).||u2||

But I am having trouble understanding what u1 and u2 should be. I understand they must be othorgonal and basis for a subspace S [itex]\in[/itex] C[-1,1].

However I am not too sure what dimension this basis should be of, and not 100% sure what is meant by the vector space C[-1,1].

(The solution uses 1 and t as u1 and u2...)

Many thanks in advance for any assistance.
 
Last edited:
Physics news on Phys.org
  • #2
I think you're making it overcomplicated. Isn't it just asking for the affine function g(t) which minimises the integral for the given f(t)?
 

Related to Inner product space - minimization.

1. What is an inner product space?

An inner product space is a vector space equipped with a special operation called an inner product, which takes in two vectors and produces a scalar value. This operation measures the angle between the two vectors and their lengths.

2. How is minimization applied in inner product spaces?

Minimization in inner product spaces involves finding the minimum value of a function defined on the space. This is done by taking the derivative of the function and setting it equal to zero, which gives the critical points. The minimum value is then found by evaluating the function at these critical points.

3. What is the relationship between minimization and orthogonality in inner product spaces?

In inner product spaces, the vectors that minimize a function are also orthogonal to each other. This means that their inner product is equal to zero. This relationship is important in applications such as least squares regression, where minimizing the sum of squared errors involves finding orthogonal vectors.

4. How is the Gram-Schmidt process used in inner product space minimization?

The Gram-Schmidt process is a method for finding an orthonormal basis for a subspace of an inner product space. This process is useful in minimization because it allows us to express any vector in the subspace as a linear combination of the orthonormal basis vectors. This simplifies the calculation of inner products and makes minimization easier.

5. Can minimization be applied in infinite-dimensional inner product spaces?

Yes, minimization can be applied in infinite-dimensional inner product spaces, but it requires some additional considerations. In this case, the derivative of the function may not exist, so we must use other methods such as the Riesz representation theorem to find the minimum value. Additionally, the concept of convergence becomes important when dealing with infinite-dimensional spaces.

Similar threads

Body forces in static equilibrium (FEM issue)
    • Mechanical Engineering
Replies
2
Views
1K
I Inner and Outer Product of the Wavefunctions
    • Quantum Physics
Replies
8
Views
2K
POTW A Modified Basis in an Inner Product Space
    • Math POTW for University Students
Replies
2
Views
887
Calculating Earth's Orbital Velocity from Varying Distance to the Sun
    • Introductory Physics Homework Help
2
Replies
42
Views
5K
Minimize integral using orthogonal basis
    • Calculus and Beyond Homework Help
Replies
2
Views
418
Question about coefficient of kinetic friction and distance
    • Introductory Physics Homework Help
Replies
4
Views
2K
I Shouldn't this definition of a metric include a square root?
    • Linear and Abstract Algebra
Replies
8
Views
1K
Null space of dual map of T = annihilator of range T
    • Calculus and Beyond Homework Help
Replies
0
Views
477
Inner Product Space and Orthogonality proof question (is this the correct way?)
    • Calculus and Beyond Homework Help
Replies
11
Views
4K
I Real function inner product space
    • Linear and Abstract Algebra
Replies
6
Views
1K

Hot Threads

Recent Insights

Back
Top