What is the meaning of T(1), T(x), and T(x2) in polynomial transformations?

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In summary, the conversation discusses a problem involving polynomial transformations and finding eigenvalues and their linear subspaces. The individual is confused about how the transformations are written and asks for clarification. The output provides a solution and also mentions the issue with the math symbols not displaying correctly in the preview. It also mentions that the topic of eigenvalues should be discussed in the Calculus & Beyond section.
  • #1
mystmyst
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Homework Statement



T: R3[x] R3[x] // for some reason the arrow symbol isn't working! When I do the arrow it previews as the third power for some reason. Also, whenever I preview post, it adds 1 2 b[3] again for some reason and I have to delete those lines every time...a bit fustrating...

T(1) = 3 + 2x +4X2
T(x) = 2 + 2x2
T(x2) = 4 + 2x 3x2

Find all eigenvalues and their linear subspaces they create.

The Attempt at a Solution



I don't exactly understand what to do here. I am used to transformations from R3 to R3, not the polynomial transformations. And I am a bit confused how they wrote T(1) T(x) and T(x2) separately. Is that the way you write the transformations for polynomials? I thought it should be like this: T(alpha0 + alpha1x + alpha2x2) = (...)

I guess I just need someone to explain what

T(1) = 3 + 2x +4X2
T(x) = 2 + 2x2
T(x2) = 4 + 2x 3x2

means.

Thanks.
 
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  • #2
T(1) = 3 + 2x + 4X^2
T(x) = 2 + 2x2
T(x^2) = 4 + 2x+ 3x^2

since they are linear:
T(a_1+a_2x+a_3x^2)=(a_1(3 + 2x + 4X^2) + a_2 (2 + 2x2) + a_3 (4 + 2x+ 3x^2))
 
  • #3
Thanks. I solved the question.

By the way, anybody have an idea why the math symbols are behaving strangely? Is this happening to anybody else?
 
  • #4
Like this?
[tex]T:R^3 \to R[/tex]

Click the expression to see the LaTeX I used.

The preview function has not been working correctly for about the past month. Instead of showing you a preview of what you have typed, it seems to take whatever is in cached memory and displays that.

Also, for future reference, problems like this should be posted in the Calculus & Beyond section. Eigenvalues are definitely not at the Precalc level.
 

FAQ: What is the meaning of T(1), T(x), and T(x2) in polynomial transformations?

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used in linear algebra to describe the behavior of a linear transformation on a vector space. An eigenvalue is a scalar that represents the amount by which an eigenvector is stretched or compressed by a linear transformation. An eigenvector is a non-zero vector that remains in its original direction after being transformed.

2. How are eigenvalues and eigenvectors calculated?

Eigenvalues and eigenvectors can be calculated by solving the characteristic equation of a square matrix. The characteristic equation is obtained by setting the determinant of the matrix minus a scalar multiple of the identity matrix equal to zero. The roots of this equation are the eigenvalues, and the corresponding eigenvectors can be found by plugging in each eigenvalue into the equation and solving for the eigenvector.

3. What is the significance of eigenvalues and eigenvectors?

Eigenvalues and eigenvectors have many applications in mathematics and science. They are used to understand the behavior of linear transformations, such as in physics and engineering. They are also used in data analysis, such as in principal component analysis, to reduce the dimensionality of a dataset while preserving the most important information. In quantum mechanics, eigenvalues and eigenvectors are used to describe the energy states of a system.

4. Can a matrix have more than one eigenvalue?

Yes, a matrix can have multiple distinct eigenvalues. In fact, the number of eigenvalues is equal to the dimension of the matrix. However, a matrix can also have repeated eigenvalues, meaning that there are multiple eigenvectors associated with the same eigenvalue. In this case, the eigenvectors may span a subspace known as the eigenspace.

5. How are eigenvalues and eigenvectors used in diagonalization?

Diagonalization is a process that transforms a matrix into a diagonal matrix using its eigenvectors. This is useful because diagonal matrices have simpler properties and are often easier to work with in calculations. To diagonalize a matrix, the eigenvectors are used to construct a matrix P, and the diagonal matrix D is obtained by calculating P^-1AP, where A is the original matrix. The diagonal entries of D are the eigenvalues of A, and the columns of P are the corresponding eigenvectors.

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