The Problem
Show that $\textrm{trace}: \textrm{End}(V)\rightarrow k$ defined by $\textrm{trace}(\phi ):=\textrm{trace}(M(\phi ))$ is a linear map.
My Question
I need some help with understanding the notation in this question. In other words, I am having a difficult time "reading the question." If anybody would be willing to dumb this down for me, I'd be incredibly grateful. I think once I understand how to start this proof, I will be be fine.
I understand:
- What $\textrm{trace}$ means.
- What $\textrm{End}(V)$ means.
- What a linear map is, and how to show that something is a linear map.
I do not understand:
- What $\phi$ means in this context.
- What $M(\phi)$ means.
- What $\textrm{trace}(M(\phi))$ means.
Additional Details
The book used is the course I am taking is Abstract Linear Algebra by Curtis, although this exercise may not be from the text itself.
As always, thank you all for your help.
Nice job framing your question precisely!
$\phi$ is the variable of this particular function. Just like the $x$ in $f(x) = x^2$. It stands for any member of $\operatorname{End}(V)$.
I don't think it's standard notation, but from the context I gather that $M(\phi)$ is a matrix associated to $\phi$. Once you choose a basis $v_1, v_2, \dots v_n$ for $V$, you know that there are constants $(a_{ij})$ such that $$ \phi(v_j) = \sum_{i=1}^n a_{ij} v_i $$ for each $j$. Then $M(\phi)$ is the matrix whose $(i,j)$th entry is $a_{ij}$.
Now you know that $M(\phi)$ is a matrix, it should be clear that $\operatorname{trace}(\phi)$ is the trace of this matrix.
Something to think about/prove: If you need to choose a basis to find $M(\phi)$, does the trace depend on that basis? If it does, trace is not a function on endomorphisms, only on matrices.
Then you have to show that the trace satisfies the linearity properties. But that should be straightforward.