[nearlynameless]

While this explanation is certainly much clearer than what I remember of high school maths, I still have a pretty tough time following the formula examples.

When I see A(x) = ax, I'm not entirely sure how to read it.

Is A meant to be a function that accepts x? If so, why is the equivalent expression a * x? Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value? Is this notation specific to multiplication, to this expression, or what?

Positing that something is 'intuitive' when it depends so much on additional contextual knowledge seems ever so slightly disingenuous as best, and slightly harmful at worst; it can make the reader feel as though they must be dumb for not understanding this 'intuitive' material.

I do acknowledge that this is linear algebra, and if one doesn't have a really solid grasp of notation of regular algebra it is likely to go over their heads, but the practical explanations (such as the slope rise/run example) are quite clear and relatively simple to follow; it follows that a simple explanation of the notation might be helpful too.

[kdtop]

I agree that this was a bit confusing. Higher up in the article, it shows that a linear function is one that doesn't change when scaled:

F(a * x) = a * F(x)

This is showing the relationship between two uses of the same function.

Then, further along, we find:

"So, what types of functions are actually linear? Plain-old scaling by a constant, or functions that look like: F(x)=ax In our roof example, a=1/3"

I think in this second situation, F(x)=ax is not a relationship but rather a DEFINITION of the function F(x).

In programming terms:

function F(x: real) : real;

begin

  Result := x * (1/3);

end;

[rustybolt]

The particular example might be confusing because the same letter is used twice with different capitalization. There is no direct relation between them.

> Is A meant to be a function that accepts x?

Yes.

> If so, why is the equivalent expression a * x?

Because that how A(x) is defined.

>Is it supposed to be implied that function A also has some hidden value "a" that is going to be multiplied by the supplied value?

Yes, it's an unspecified constant (a, b, c... are used to denote constants by convention), so you can really calculate A(x) for supplied values of x yet until the constant 'a' is specified.

> Is this notation specific to multiplication, to this expression, or what?

No. Functions might be defined using any expression. For example

    A(x) = b^x

is a valid function as well (again, we have an unspecified constant). Just don't expect to encounter it in an introductory course in linear algebra (since it would deal mostly with linear functions).