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

[mystmyst]

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.

 

[justsof]

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))

 

[mystmyst]

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?

 

[Mark44]

Like this?
$$T:R^3 \to R$$

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.