Proving Trace is a Linear Functional

Proof

1. We define $trace(A)={\sum }_{n=1}^{\infty }{A}_{ii}$
2. By definition, $trace:{\mathbb{F}}^{n\times n}\to \mathbb{F}$
3. Thus, if we show trace is a linear linear transformation, $trace\in L({\mathbb{F}}^{n\times n},\mathbb{F})$

Proving Linear Transformation

1. Let $c\in \mathbb{F},A,B\in {\mathbb{F}}^{n\times n}$
2. We denote $A$ by $A_{ij}$ and $B$ with $B_{ij}$
3. Then, $trace(cA+B)={\sum }_{n=1}^{\infty }(cA+B{)}_{ii}$
4. $={\sum }_{n=1}^{\infty }(cA{)}_{ii}+B_{ii}$ by defn of matrix addition
5. $={\sum }_{n=1}^{\infty }c(A_{ii})+B_{ii}$ by defn of scaling matrixes
6. $=c({\sum }_{n=1}^{\infty }{A}_{ii})+{\sum }_{n=1}^{\infty }(B_{ii})$ by field properties, asocciative
7. $=c\ast trace(A)+trace(B)$
8. Thus, it follows that $trace$ is a Linear Function
9. Thus, it follows that $trace$ is a Linear Functional