Hi, I'm Kalish. I have a problem and a proposed solution. Please point out any errors!

Not sure about this one. Here's my work for it:

Problem Statement: Determine the dimensions of the kernel and the image of the linear operator T on the space R^n defined by T(x_1,...,x_n)^t=(x_1+x_n,x_2+x_{n-1},...,x_n+x_1)^t.

My attempt: The dimension of the kernel is the number of vectors in the basis for [0,0,...,0]^t, or 0, because the dimension is 0. The dimension of the image = 1, because by the dimension formula, dim(image(T))=dim(V)-dim(ker T)=1-0=1.

This seems too easy to be true. Any thoughts? Thanks.