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).
> 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
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).