@mmcdara The free and generating families are useful notions. Usually it is simpler to check these two, instead of a single condition. E.g., if a family F is free in a vector space E, then F is a basis for the vector subspace generated by F in E.

@Christopher2222  The strange fact is that evalb and select do not confirm numboccur.
numboccur(L, 1.) = nops(select(`=`, L, 1.));  # 2 = 3

@Carl Love Yes, of course.

Q:=eval([x,y], solve([x+2*y-1, 2*(x-6) - (y+3)])): # projection onto directrix
F:=2*[6,-3]-Q:
P:=(x-F[1])^2 + (y-F[2])^2 = (x+2*y-1)^2 / 5:
with(plots): display(implicitplot([P, x+2*y-1], x=-10..10,y=-10..10), pointplot([[6,-3], F], color=[red,blue]));

@JAMET You have already an answer. What did you not understand?

@mmcdara The unicity (uniqueness) of the linear combination makes the family free. So, the definitions are the same.

@Carl Love Vote up (especially for your comeback)!

@tomleslie The .eps is correct (with a few artifacts though). ghostscript converts it properly into a vector graphics pdf.
Maybe GIMP has an option to do that too.

@tomleslie Your file test2.pdf obviously contains a low resolution bitmap. Have you zoomed into it?

Both your graphics are bitmaps (not vectorized).
To obtain true vector graphics, convert the eps to pdf using e.g. ghostscript or latex.
Compare the results by zooming in:

from-eps.pdf

@Anthrazit Unlike sin or arctan, arctand does not check the (number of) arguments. You may use:

arctandeg := () -> 180/Pi*arctan(args);

@mmcdara Using changes of variables it is easy to integrate f (just like K above). This actually means that the user must know to compute it by hand in order to use Maple (!).

@max125 In a vector space, you can add two vectors and multiply a scalar by a vector. But adding a vector and a scalar is a nonsense (however, in Maple, for matrices, a scalar is identified with a scalar matrix).

3 + <1,2,3>;
Error, a constant cannot be added to a Vector; use +~ for elementwise addition instead of +
The same convention was implemented for lists, but here instead of an error, the expression is returned unsimplified (just like 3 + "abc")

Note that map(f, 3, L)  ==> f(3, L), and for f = `+`, the above remark applies.

@jeffreyrdavis75 It is not clear (at least for me) whether you have some mathematical problems, or you just want some nice pictures.

@mmcdara  Sorry for the typo  :-)