Linear Transformations: Find Eigenvalues & Eigenvectors

[mivanova]

Please, help me!
Suppose n is a positive integer and T is in F^n is defined by
T(z_1, z_2, ... , z_n) = (z_1+ ... +z_n, z_1+ ... +z_n, ...,z_1+ ... +z_n)
Determine all eigenvalues and eigenvectors of T.
Thank you in advance!

 

[HallsofIvy]

Thanks in advance for what? Telling you the answer? No, that's not going to happen.

Making suggesteins and giving you hints? Hard to do without knowing what you DO know about this problem. And since you have shown nothing at all about what you have tried, we can't know that. Do you, for example, know how to write T as a matrix? Could you find the eigenvalues and eigenvectors then? Since all of the components of the T(z) are the same, no matter what z is, at least one eigenvalue should be obvious. What is T(1, -1, 0, 0, ..., 0)?

I am wondering what, exactly, you mean by "T is in F^n". My first guess would be that F^n is the set of ordered n-tuples of some field F, with component wise addition and scalar multiplication. But in that case it is (z_1, z_2, ..., z_n) that is "in F^n", not T. T is in L(F^n, F^n).