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with(Groebner):
T := lexdeg([x,y,z],[e1,e2]);
intermsof1 := y;
intermsof2 := -z;
GB := Basis([e1-intermsof1, e2-intermsof2], 'tord',T);
result := NormalForm(y^2-x*z, GB,'tord', T);
result := NormalForm(y^2-x*z, GB, T);

originally Basis do not have error when without parameter 'tord'

after it has argument error, it has to be added extra parameter tord

NormalForm has the same error too.

i do not understand why it has error, how to solve?

i just want to express y^2-x*z in terms of y and -z

 

 

How to generate all basis of a set? (rather an the one basis that basis generates)

how to get maple to do linear algebra in Z_2 (integers modulo 2)

I don't want it to solve and then reduce mod 2 I want it to work over Z_2 so basis([ [1,1,1], [1,-1,1 ]) = [1,1,1]  etc


Hi all,

I am considering a scenario in which I have, for example, four matrices, A, B, C, and D, which form a basis for all of the (numerical) calculations I am doing.  (For example, A + B = i*C, etc.)  Right now, if I add A and B, I get a matrix back whose elements are i*C, but I cannot get Maple to express it as i*C.  As a simple example, let:

A = (1 0 // 0 1 )

B = (0 1 // -1 0)

C = (i i // -i i)

Then A + B returns (1 1 // -1 1); I'd like for Maple to "intelligently" give iC.  So...how can I get Maple to expand a given matrix (A+B) in terms of a particular basis (here, simply C)?

Thank you.

 

g:=Groebner:-Basis([a-2.0*b,b-2], plex);

Groebner:-Reduce(a, g, plex); 

Error, (in content/polynom) general case of floats not handled

How to solve this problem simply?

 

i use a not good example's polynomials to illustrate the idea

u1:=3*a*c+b^2+c;
u2:=7*a^5*b*c^3;
u3:=a+3*b^2;
T := lexdeg([a,b,c],[e1,e2, e3]);
GB := Basis([e1-u1, e2-u2, e3-u3],T);
result := NormalForm(a*b+b*c, GB, T);

now result is to express a*b+b*c in terms of e1, e2, e3 which represent u1, u2, u3.

is it possible to use preimage to find possible u1,u2,u3 if unknown u1,u2,u3 and given known eqx?

How to use preimage to find possible eqx if given known u1,u2,u3 to find eqx in NormalForm(eqx, GB, T)?

 

what i confused in code below is that if i know it in terms of -1*e1+2*e2+*e3

it already can be used to find eqx, it seems reasonable to put source1list as unknown to find eqx  or find unknown eq1, eq2, eq3 if given known eqx.

with(RegularChains):
with(ConstructibleSetTools):
source1 := PolynomialRing([e1, e2, e3]);
target1 := PolynomialRing([a, b, c]);
source1list := [...];
target1list := [eq1, eq2, eq3];
cs := PolynomialMapPreimage(target1list, source1list, source1, target1);
Info(cs, source1);
[[e1-e2-e3], [1]]
-1*eq1+2*eq2+*eq3;

follow Computing non-commutative Groebner bases and Groebner bases for modules

in maple 12

Error, (in Groebner:-Basis) the first argument must be a list or set of polynomials or a PolynomialIdeal

 

then i find in maple 15 help file is changed from module M := [seq(Vector(subsop(i+1 = 1, [F[i], 0, 0, 0])), i = 1 .. 3)]

to array M := [seq( s^3*F[i] + s^(3-i), i=1..3)];

though it can run, but when apply other example can not run

such as

 

restart;
with(Groebner):
F := [2*x^2+3*y+z^2, x^2*z^2+z+2*x, x^4*y^7+3*x];
M := [seq( s^3*F[i] + s^(3-i), i=1..3)];
with(Ore_algebra);
A := poly_algebra(x,y,z,s);
T := MonomialOrder(A, lexdeg([s], [x,y,z]), {s});
G := Groebner[Basis](M, T);
Error, (in Groebner:-Basis) the first argument must be a list or set of polynomials or a PolynomialIdeal

G1 := select(proc(a) evalb(degree(a,s)=3) end proc, G);
[seq(Vector([seq(coeff(j,s,3-i), i=0..3)]), j=G1)];
C := Matrix([seq([seq(coeff(j,s,3-i), i=1..3)], j=G1)]);
GB := map(expand, convert(C.Vector(F), list));
Groebner[Basis](F, tdeg(x,y,z));

#page 320 and 322 of book Singular introduction to commutative algebra

it return too many recursion 

 

hilbertseries([a+a*c, a+a*b, a+b+c]);

eq1 := a+a*c;

eq2 := a+a*b;

eq3 := a+b+c;

eq1a := Homogenize(eq1, h);

eq2a := Homogenize(eq2, h);

eq3a := Homogenize(eq3, h);

T3:=lexdeg([a,b,c,h]);

GB := Basis([eq1a,eq2a,eq3a], T3); #a

 

#MonomialHilbertPoincare(LeadingMonomial(GB[1],T3), LeadingMonomial(GB[2],T3), LeadingMonomial(GB[3],T3));

 

with(PolynomialIdeals):

MonomialHilbertPoincare := proc (I3)

#I3:=[LeadingMonomial(GB[1],T3), LeadingMonomial(GB[2],T3), LeadingMonomial(GB[3],T3)];

T2:=lexdeg([h,c,b,a]);

varj := [h,c,b,a];

I2 := InterReduce(I3, T2);

s := nops(I2);

if I2[1] = 0 then return 1 end if:

if I2[1] = 1 then return 0 end if:

if degree(I2[s]) = 1 then return (1-varj[1])^s end if:

lt := LeadingTerm(I2[s],T2);

leadexp := [degree(lt[2],h),degree(lt[2],c),degree(lt[2],b),degree(lt[2],a)];

j := 1;

for z from 1 to nops(leadexp) do

                if leadexp[j] = 0 then

                                j := j + 1;

                end if:

od:

finallist := [];

for z from 1 to nops(GB) do

                finallist := [op(finallist), GB[z]+varj[j]];

od:

quotientlist := Generators(Quotient(GB, varj[j]));

finallist2 := [];

for z from 1 to nops(quotientlist) do

                finallist2 := [op(finallist2), op(z,quotientlist)];

od:

return MonomialHilbertPoincare(finallist) + varj[1]*MonomialHilbertPoincare(finallist2);

end proc;

F:=[LeadingMonomial(GB[1],T3), LeadingMonomial(GB[2],T3), LeadingMonomial(GB[3],T3)];

MonomialHilbertPoincare(F);

 

Let Poly2 denote the vector space of polynomials

(with real coefficients) of degree less than 3.

Poly2 = {a1t^2+ a2 t+ a3 |a1; a2; a3 €R}

You may assume that {1,t; t^2}is a basis for Poly2.

(1) Show that L1 = {t^2 + 1; t-2 ; t + 3}and L2 = {2 t^2 + t; t^2 + 3; t}

are bases for Poly2.

(2) Let = 8t^2- 4+ 6 and = 7t^2- t + 9. Find the coordinates for

and with respect to the basis L1 and with respect to the basis L2

(3) find the coordinate change matrix P from the basis L1 to the basis L2.find P^-1

Just I answer part (1) can you help me to answer 2 and 3 

 V = {(2a + b + c, 3a + b + 2c, 2a + c, b + 7c)|a, b, c ∈ R} forms a subspace of R^4 
 
 Show that {(2, 3, 2, 0),(1, 1, 0, 1),(0, 1, 1, 6)} is an (ordered) basis B for V . 

toally confused on how to do this??..please help!!

I am trying to take out the intersection between two bases from one of the original basis. I have two matrices (A and B) and want to find the intesection between the range (or column space) of A and the null space of B. The range of A is 

X1:=ColumnSpace(A)

For some reason, the column space is not presented as a list, but with square brakes, so I convert this into a list of vectors:

X2:=SumBasis([X1[1],X1[2]])    %If I use the command Basis, it returns again square brakets, not sure why... 

The null space of B

X3:=NullSpace(B) 

X4:=IntersectionBasis([X2,X3])

X5:= X2\cap X4 % I am using latex code for the intersection symbol...

The result is the empty set! Evidently, X2 and X3 are differnt bases.

Any help would be really welcome! Many thanks!

Joaquin

So im trying to write a maple script that computes the Jordan form of a given (3x3)- matrix
A. If {a,b,c} is a basis with respect to which A is in Jordan form, then I'm trying to make it
plot the three lines spanned by a, b and c, in the standard coordinate system. I was hinted to use plot3d here.

sidenote: I know how to compute the jordan matrix of A, such by find the eigen vectors and generalised eigen vectors and putting them in as columns in a 3x3 matrix say S,   where S is invertible    then  (S^-1)*(A)*(S) = (J).

Thanks in advance. <3

In that attached file is a multip step problem that involves graphing a right circular cylinder using transtion matrices and orthonormal basis. I have completed the hole question minus the very last part which is asking for new parametric equations for the cylinder if its center point is located at (-2, 10, 3) instead of the origin.

Any ideas on how to do this will be greatly appreciated.

 

question_4.mw

Find parametric equations for the right circular cylinder having radius 3, length 12, whose axis is the z-axis and whose bottom edge lies in the plane: z=0.

 

 

Do I just define B={u1, u2, u3} being a basis for R3 and use the gram-schmidt operator to find the parametric equations?

I know that would give me an orthonrmal basis, but how do i find parametirc equations?

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