Vertical Section of Linear Combination of Functions is Linear Combination of Vertical Sections
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Theorem
Let $X$ and $Y$ be sets.
Let $f_1, f_2, \ldots, f_n : X \times Y \to \overline \R$ be functions.
Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be real numbers.
Let $x \in X$.
Then:
- $\ds \paren {\sum_{k \mathop = 1}^n \alpha_k f_k}_x = \sum_{k \mathop = 1}^n \alpha_k \paren {f_k}_x$
where $f_x$ denotes the $x$-vertical section of the function $f$.
Proof
Let $y \in Y$.
We have:
\(\ds \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k}_x} y\) | \(=\) | \(\ds \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k} } {x, y}\) | Definition of Vertical Section of Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \alpha_k \map {f_k} {x, y}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^n \alpha_k \map {\paren {f_k}_x} y\) | Definition of Vertical Section of Function |
so:
- $\ds \paren {\sum_{k \mathop = 1}^n \alpha_k f_k}_x = \sum_{k \mathop = 1}^n \alpha_k \paren {f_k}_x$
$\blacksquare$