Find the Kernel of the Trace of a Matrix

[nautolian]

Homework Statement

Let F : M_nn(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?

Homework Equations

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 M_nn}? And would the Im(F) just be {a : a\inℝ}? Thanks.

 

[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.

 

[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