Proving Orthonormal Basis & Uniqueness of Inner Product | Linear Algebra HW Help

In summary, the conversation discusses proving that any basis of R^n is an orthonormal basis with respect to some inner product and whether the inner product is uniquely determined. The conversation also touches on the definition of inner products in R^n and how symmetric matrices can induce inner products. The concept of uniqueness is clarified as there being only one inner product that satisfies the properties of being an orthonormal set. There is some confusion about the specific notation and formula for the inner product.
  • #1
blue2004STi
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Homework Statement


Prove that any basis of R^n is an orthonormal basis with respect to some inner product. Is the inner product uniquely determined?


Homework Equations


I am not sure where to begin. Should I just define an arbitrary basis for a arbitrary R^n? I mean I think I understand the question about the inner product being uniquely determined but I am not sure where to begin.


The Attempt at a Solution


See above.
 
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  • #2
How do you define inner products in R^n? A familiar question: how and when do symmetric matrices induce inner products on R^n?

Uniquely determined means that there is no other inner product that has those properties. That is, given a basis there is only one inner product that makes the basis an orthonormal set. If you don't know what I meant by symmetric matrices you can just play around with scaling inner products by positive reals.
 
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  • #3
Do you mean in (x^T)Kx or in notation <x,x> or in formula? I'm not going to lie I'm a bit confused with what you're asking.
 
  • #4
Do you mean by bilinearity, symmetry, and positivity?
 

FAQ: Proving Orthonormal Basis & Uniqueness of Inner Product | Linear Algebra HW Help

What does it mean for a basis to be orthonormal?

An orthonormal basis is a set of vectors in a vector space that are mutually orthogonal (perpendicular) and have a unit length. This means that all the vectors are independent of each other and have a magnitude of 1.

How do you prove that a basis is orthonormal?

To prove that a basis is orthonormal, you need to show that the vectors are orthogonal (their dot product is equal to 0) and that each vector has a magnitude of 1. This can be done using the Gram-Schmidt process or by directly calculating the dot products and magnitudes of the vectors.

What is the inner product in linear algebra?

The inner product in linear algebra is a mathematical operation that takes two vectors and produces a scalar (number) as the result. It is also known as the dot product and is calculated by multiplying the corresponding components of the two vectors and then adding them together.

How do you prove the uniqueness of an inner product?

The uniqueness of an inner product can be proven by showing that it satisfies the four axioms of an inner product: linearity, positive definiteness, conjugate symmetry, and additivity. If these axioms are satisfied, then the inner product is unique.

Why is it important to prove the orthonormality and uniqueness of an inner product?

Proving the orthonormality of a basis and the uniqueness of an inner product ensures that the vectors being used are independent of each other and that the inner product is well-defined. This is crucial in many applications of linear algebra, such as solving systems of equations and finding eigenvalues and eigenvectors.

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