At the first note that in this question all polynomials have parametric coefficients. Let F be a list of polynomials and f be a polynomial. I want to convert F and f into a linear homogeneous FF and ff resp. At the first I want to sortvthe monomials appears in F and f  w.r.t. a monomial order T and then replace by the new variables A_i.

For example if

and

(a,b,c are parameters and x,y,z are variables) then I want to convert F and f into FF and ff resp:

please note that the variables appears in F and f are:

where sorted by T=plex(x,y,z). Please note that we consider all constants and alone parameters (4, b-4, c-1) as A9. I want to convert v into

and then F into FF and f into ff.

How can I extract the coefficients of all monomials in a multivariate polynomial?

For example if f=ax^2+bxy^3+2 then I want

coeff(f,x^2)=a

coeff(f,xy^3)=b

coeff(f,1/2)=4

coeff(f,1/10)=20 and...

I know the Wronskian command. I want to use this command for detecting linearly dependence or independence of some polynomials. I know that the polynomials  are  linearly independent if the Wronskian is not zero.  Conversely, if the Wronskian vanishes  then the polynomials are linearly dependent. Now I want to know that how can I find the coefficient vector if the polynomials are dependent by Wronskian command?

For example if [f1,f2,f3] be a list of polynomials s.t. a1f1+a2f2+a3f3=0. How can I find a1, a2, a3? 

How  we can decide whether a polynomial f is a linear combination of some polynomials g1,...,gm?

For example if f=x^2+y^2 and g1=y+x^2 , g2=y^2-y then f=g1+g2.

Let M be the matrix [[]] means that a matrix without any array. If we call RowDimension(M) then it output is 1 and we know that it is zero. What I have to do? It is hilarious that the RowDimension([[1]])=RowDimension([[]])!!!