# Thread: Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One

1. ## Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One

See attachments for problem and note that I want to solve the problem for n=2 only. Also, my attempted solution is attached as well.

Question:

N(T) denotes the nullspace of T. I found the nullspace, but I'm not sure if I have the correct basis for it or if I have written the solution space with the correct notation. Obviously, the nullspace will be the set of vectors where the 2 diagonal components of the matrix are additive inverses of one another. But given that either of the diagonal components could be negative, how do I write this in proper format?

Also, did I get the correct basis?

Thanks for your time.

2. ## Re: Trace(A) - Finding Nullspace and Range; Deciding if Onto and/or One-to-One

We have $T:M_{n\times n}(F)\to F$ given by $T(A)=\textrm{tr}(A)$ . For $n=2$ , $A=\begin{bmatrix}{x}&{y}\\{z}&{t}\end{bmatrix}\in N(T)$ iff $x+t=0$ so, $A\in N(T)$ iff $A$ has the form:

with $x,y,z\in F$. Easily proved, those three matrices are linearly independent and as a consequence form a basis for $N(T)$ .