Similarity and change of basis

In summary, The conversation discusses finding the matrices [T]B and [T]E of a linear operator T on a 2-dimensional complex vector space V, defined by T(f(x)) = f(x + α), relative to the bases B = {cos x, sin x} and E = {cos x + i sin x, cos x − i sin x}. The task is to find an invertible matrix P that implements the similarity between [T]B and [T]E. The speaker has already found T_B and T_E, but needs help with finding P. The other speaker gives a hint about using knowledge of transformation matrices between different bases.
  • #1
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Homework Statement



Consider the 2-dimensional complex vector space V of functions spanned by sin x and cos x. For a fixed real number α, define a linear operator T ≡ Tα on V by putting T(f(x)) = f(x + α). Find the matrices [T]B and [T]E of T relative to the bases B = {cos x, sin x} and E = {cos x + i sin x, cos x − i sin x}. Find an invertible matrix P implementing the similarity between [T ]B and [T]E :

[tex][T]_B = P[T]_EP^{-1}[/tex]

Homework Equations





The Attempt at a Solution



I found T_B and T_E and I think I'm probably right because their tr and det are equal. Sorry if I don't show all that work because it is long. My problem is to get P. I am not sure how to get this. Thanks for any suggestions.
 
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  • #2
Well, what do you know about transformation matrices between different bases?
 
  • #3
OK. Got it now.
 

1. What is similarity in the context of linear algebra?

Similarity in linear algebra refers to the relationship between two matrices that have the same properties, such as determinant, eigenvalues, and trace. These matrices are said to be similar if they can be transformed into each other through a change of basis.

2. What is a change of basis in linear algebra?

A change of basis is a mathematical operation that transforms a vector or a matrix from one coordinate system to another. It involves finding a new set of basis vectors that can represent the same vector or matrix in a different way.

3. How is similarity related to change of basis?

Similarity and change of basis are closely related because a change of basis is necessary to determine if two matrices are similar. By transforming both matrices into the same basis, we can compare their properties and determine if they are similar.

4. Can two matrices be similar but have different dimensions?

Yes, two matrices can be similar even if they have different dimensions. This is because similarity is based on the properties of the matrices, not their size or shape. As long as the matrices have the same determinant, eigenvalues, and trace, they can be considered similar.

5. What is the significance of similarity and change of basis in real-world applications?

Similarity and change of basis are important concepts in linear algebra that have many practical applications. They are used in fields such as physics, engineering, and computer science to simplify calculations, analyze data, and solve problems involving transformations of data or systems. These concepts also play a crucial role in the understanding of geometric transformations and the behavior of linear systems.

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