Matrix of a linear transformation HELP

[5kold]

Homework Statement

Show that if the matrix of a linear transformation
"multiplication by a" is "A" then a is a root of the characteristic polynomial for A.

Also, I am not sure how to obtain the monic polynomial of degree 3 satisfied by
2^(1/3) and by 1+2^(1/3)+4^(1/3).

The Attempt at a Solution

It seems obvious but I am not sure how to go about it. I can't find anything in my book about it. Once you have the matrix form, can't you just plug in the numbers above to get the polynomial equation?

Thank you guys. This is my first course in higher mathematics.

 

[Mark44]

Do you know what the matrix A looks like? See what T does to each vector in the standard basis.

Do you know how to find the characteristic polynomial for a matrix A?

 

[5kold]

I do not know what the matrix A looks like. What is T? I do know how to find the characteristic polynomial for a square matrix. Thanks for the continuing help, Mark44.

 

[Mark44]

T(x) = ax, right? What does T do to each basis vector of the standard basis? From that you should be able to figure out what A looks like.

Just to help you understand better you can temporarily assume that T takes a vector from R³ and maps it to another vector in R³. After you understand what's going on, then you should assume that T maps vectors in Rⁿ to Rⁿ. I think that's a reasonable assumption for this problem.