linear algebra foundation of Riemann integrals Let $V$ be the vector space of real functions $f\colon [a,b]\to \mathbb R$ and let $X$ be the set of characteristic (indicatrix) functions of subintervals: $X=\{\mathbb 1_I\colon I\subset [a,b] $ interval $\}$. We define $T\colon X \to \mathbb R$ as $T(\mathbb 1_I) = |I|$ where $|I|$ is the length of the interval $I$. Notice that $X$ is not a set of independent vectors because the sum of adjacent intervals is again an interval, but in that case $T$ is defined to be additive.
So it is clear that $T$ can be extended as a linear map on the vector space generated by $X$ (which is the space of so called simple functions). 
What are the abstract properties of $X$ and $T$ (in the setting of linear algebra) which can be applied to the above example to prove that $T$ is linear on $X$?
For example $X$ and $T$ have the following property: $x,y,x+y\in X \implies T(x+y) = T(x)+T(y)$ and $x,\lambda x \in X \implies \lambda=1$. Is this enough to prove that $T$ is linear i.e. that $x,y,\lambda x +\mu y \in X \implies T(\lambda x + \mu y) = \lambda T(x) + \mu T(y)$? And is this enough to prove that $T$ has a linear extension to the span of $X$?

I think the only thing that needs to be proved for $T$ to be extendable to a linear map on the vector space generated by $X$ is the following. Assume $f=\mathbb{1}_{[c,d)}$ and $g = \mathbb{1}_{[d,e)}$ with $f,g \in X$. Then $f+g$ is also in $X.$ (I am assuming that half-open intervals are used here.) It is then needed that $T(f+g) = T(f) + T(g)$ for all such $f$ and $g$ in $X.$ 
This is because if the above is true (which it is!), then you can show that any function that can be written as a linear combination of elements of $X$ (i.e. a simple function) maps to a unique value by the extension of $T$, regardless of how it is written as a linear combination.
