How to shaw that any element of R² is a linear combination of (1,1) and (1,2), this decomposition is unique. The family is free and generates R². Thank you.

How to know if 2 pairs of vectors generate the same vector space ?
The 2 pairs of vectors are <1,0-1>  and <0,1,0> for une part, <2,1,-2> and <-1,2,,1> for the other. Thank you.

Sorry, I don't understand what are M, V, and sys.
How to get a base line of the linear application image ? Thank you.

Iam trying my own procedure taylor on sin(t) for t=0

f := t -> diff(sin(t), t);
f := proc (t) options operator, arrow; diff(sin(t), t) end proc

L := [seq(diff(f(t), [t $ i]), i = 1 .. 9)];
L := [-sin(t), -cos(t), sin(t), cos(t), -sin(t), -cos(t), sin(t), 

  cos(t), -sin(t)]

t := 0;
L;
                             t := 0

                 [0, -1, 0, 1, 0, -1, 0, 1, 0]

for k to 8 do
    -L[k + 1]*'t'^k/k!;
end do;
                               t

                               0

                               1  3
                             - - t 
                               6   

                               0

                              1   5
                             --- t 
                             120   

                               0

                              1    7
                           - ---- t 
                             5040   

                               0
How to get the sum ? Thank you very much.

On the model : t := 't';
taylor(sin(t), t = 0, 9);
                  1  3    1   5    1    7    / 9\
              t - - t  + --- t  - ---- t  + O\t /
                  6      120      5040           

with(LibraryTools);
LibLocation := cat(kernelopts(mapledir), "/lib/myLib.mla");
   LibLocation := "C:\Program Files\Maple 2018/lib/myLib.mla"
Create(LibLocation);
Error, (in march) archive data file could not be created
a := 10;
Save('a', LibLocation);
Error, (in LibraryTools:-Save) could not open `C:\Program Files\Maple 2018/lib/myLib.mla\a.m` for
Why this Library can't be created ? Thank ypo.

I get these messages. How to correct them ? Thank you.
Error, (in march) archive data file could not be created
Error, (in LibraryTools:-Save) could not open `C:/tmp/test.mla\MyPackage.m` for writing
Error, (in march) there is no existing archive in "C:/tmp/test.mla"
Error, invalid input: with expects its 1st argument, pname, to be of type {`module`, package}, but received MyPackage

My directory is Système(C)\ Program Files\Maple 2018\lib, so I have this message in the file Filters.mw: 
libname;
               "C:\Program Files\Maple 2018\lib"
LibraryTools:-Save(Filters,cat(libname[1],"/Filters.mla"));
Error, (in LibraryTools:-Save) could not open `C/Filters.mla\Filters.m` for writing
Is it possible to overcome this problem ? Thank you.

How to change this code  drawLines3.mw  to work with maple 18. Thank you very muche.

Thank you for your answer. Is it possible showing without geom3d ?

Sorry for my bad explications. I wish the 3 lines a, b, c and their projections on the same ploting. Thank you.

how to make a fold as in descriptive geometry and represent the 3 lines in the 2 planes O,x,z and O,x,-z as well as the ground line; Is it possible ? Thank you very much.

how to avoid confusion between sol1 and 2 in Fig

Fig := proc (k)
local theta, Cor, corde, P, tgP, sol, tgP1, TgP, TgP1;
global a, b, ell, Lieu; theta := k;
corde := a*x/cos(theta)-b*y/sin(theta) = a^2-b^2;
P := [a*cos(theta), b*sin(theta)];
tgP := x*P[1]/a^2+y*P[2]/b^2 = 1;
sol := solve({ell, corde}, {x, y}, explicit);
tgP1 := x*rhs(sol[2][1])/a^2+y*rhs(sol[2][2])/b^2 = 1;#there is confusion between sol[1)and sol[2]
Cor := implicitplot(corde, x = -a .. a, y = -b-5 .. b+5, color = blue);
TgP := implicitplot(tgP, x = -15 .. 15, y = -15 .. 15, color = green);
TgP1 := implicitplot(tgP1, x = -15 .. 15, y = -15 .. 15, color = magenta);
display([Ell, Cor, Lieu, TgP, TgP1], axes = normal, view = [-15 .. 15, -10 .. 10], scaling = constrained) end proc;
Fig(Pi/6):
nframes := 30; 
plots:-display([seq(Fig(2*Pi*i/nframes+0.001), i = 1 .. nframes)], insequence, scaling = constrained)
Thank you for your answer.

nframes := 100; plots:-display([seq(Fig(2*Pi*i/nframes), i = 0 .. nframes)], insequence, scaling = constrained);
There is still an error in the animation, I can't correct.
Error, (in factor/fac1mod/DegreeSet) unable to evaluate sign

Sorry your suggestion doen't work restart; _local(D); A := [-L, 0]; B := [L, 0]; C := [x, y]; D := [-x, y]; Dist := proc (X, Y) options operator, arrow; sqrt((X[1]-Y[1])^2+(X[2]-Y[2])^2) end proc; Eq := (Dist(C, D) = Dist(C, B))^2; centre := [solve(diff(Eq, x), x), solve(diff(Eq, y), y)]; with(geometry); `assuming`([conic(p, Eq, [x, y])], [L > 0]); detail(p); asymptotes(p); y_acymp := `~`[solve](`~`[Equation](asymptotes(p)), y); y := solve(Eq, y)[1]; P := proc (X, L0) local Curve, Asymptote, Trapezoid, T, pt, Ip; Curve := plot(eval([y, -y], L = L0), x = -(1/3)*L0 .. 15, color = red, thickness = 3); Asymptote := plot(eval([-sqrt(3)*x-(1/3)*sqrt(3)*L, sqrt(3)*x+(1/3)*sqrt(3)*L], L = L0), x = -(1/3)*L0 .. 15, linestyle = 3, color = black, thickness = 0); Trapezoid := plottools:-polygon(eval([A, B, C, D], [L = L0, x = X]), color = "LightGreen"); Ip := `~`[evalf]((1/3)*(eval(C, [x = X-(1/3)*L0, L = L0]))); printf("%a", Ip); T := plots:-textplot([[(eval(A, [x = X, L = L0]))[], "A"], [(eval(B, [x = X, L = L0]))[], "B"], [(eval(C, [x = X, L = L0]))[], "C"], [(eval(D, [x = X, L = L0]))[], "D"], [Ip[], "I"]], align = {above, left}, font = [TIMES, 16]); pt := plot([A, B, C, Ip], style = plottools:-point, color = blue, symbolsize = 15); plots:-display(Curve, Asymptote, Trapezoid, T, pt, scaling = constrained, size = [400, 800]) end proc; a := 7; P((1/2)*a, 6)