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I was watching a video on functional programming and the speaker was introducing functional programming notation for functions:

". . . if g(a) is a function g with variable a, we can write it also as g a . . . now, this looks like g times a and this would still be true if g were a linear equation . . ."

What does he mean by this? Why is g(a) the same as g times a if g is a linear equation? Or am I misinterpreting him? Here's the link: (Starting at ~12:30)

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Recall that a function $f$ is called linear if it verifies $f(x+y)=f(x)+f(y)$ and $f(\lambda x)=\lambda f(x)$ for all $x,y$ in its domain of definition and $\lambda \in \mathbb{R}$.

It turns out that the only linear functions on $\mathbb{R}$ are the $f(x)=ax$, where $a$ is a constant : in other words, in one dimension, linear functions are multiplication functions.

In higher dimensions though, there are a lot of different linear functions, but we tend to still use the notation $fx$ instead of $f(x)$. For one thing, it is coherent with the definition of linearity : $f(x+y)=fx+fy$ and $f(\lambda x)=\lambda (fx)$. So in a way you can think of it as a sort of "generalized multiplication", the great difference being that division in in general not allowed.

Addendum : the term of "equation" in inappropriate in this context, I guess the author meant function, or maybe expression.

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Linear functions are often represented by matrices, in which case function application is the the same as multiplication of a matrix by a vector. If $f$ is a linear function, say, from $\mathbb{R}^m$ to $\mathbb{R}^n$, then there is an $n \times m$ matrix $A$ representing the function $f$ so that for any $v$ in $\mathbb{R}^m$, $f(v) = A v$. (For the multiplication, regard $v$ as a column vector with $m$ entries.)

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