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What about a linear combination of linear functions?

Let $f,g,\dots :\mathbb R \to \mathbb R$ be a family of linear functions. We define a linear function $h:\mathbb R \to \mathbb R$ as
$$
h(x)=\sum_{i=1}^n k_i f(x_i)
$$
where $x=(x_1,x_2,\dots,x_n)\in\mathbb R^n$ and $k_i\in\mathbb R$ for all $i=1,2,\dots,n$. It appears that $h$ is also a linear function. Is this correct? If so, how do we show that it is a linear function?
I can show it using the definition of a linear function $\sum_{i=1}^n k_i (ax_i+b)=\sum_{i=1}^n k_i a x_i +b$ but I don’t know whether I can generalise it.

A:

Yes. In particular, a linear combination $f(x_1)+g(x_2)+\cdots +h(x_n)$ is a linear combination of linear functions since
\begin{align}
h(x_1)+h(x_2)+\cdots+h(x_n)&
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