# Find the Kernel of the Trace of a Matrix

1. Oct 15, 2012

### nautolian

1. The problem statement, all variables and given/known data

Let F : Mnn(R) → R where F(A) =tr(A). Show that F is a linear transformation. Find the kernel of F as well as its dimension. What is the image of F?

2. Relevant equations

3. The attempt at a solution

I have shown that it is a linear transformation. But I am not sure about the Ker(F) and Im(F),
would Ker(F) just be {tr(A), for A in Mnn}? And would the Im(F) just be {a : a$\in$ℝ}? Thanks.

2. Oct 15, 2012

### HallsofIvy

Do you understand what the problem is asking? You are gvien a linear transformation that maps every matrix to a number, its trace. This problem is asking for the trace of that linear transforamation- the set of matrices that are mapped to 0. It is asking for a set of matrices, not a set of numbers.

3. Oct 15, 2012

### nautolian

Okay, so would I say kernel is {A where tr(A)=0, for A in Mnn}? So what would the image be then? Thanks