Automatically create matrix with changing function

Hi all,

I am trying to solve the following problem:

For some analysis using the variable "xi" I need to construct a vector. In this vector, every entry is a function of xi^k, in which k is the row number and xi must stay a variable.

Now, I could just enter this by hand as a matrix, however, would it not that I need about 100 rows. Yeah, that is too time consuming.

So, I am looking for a method to construct this matrix automatically but have not succeeded yet. 

What I want is some function/program that, when putting in the amount of rows (k) and the function of xi f(xi,k), it constructs a vector with xi as a variable and k evaluated depending on the row position. I attached the file for clarification.

Do you know a solution?

Thanks a lot!