# converting linear map to matrix representation

I have a linear map $\alpha$ from $F_{p^n} \longrightarrow F_{p^n}$, where we see $F_{p^n}$ as a vector space over $F_p$ with a $V_i$ as base elements. I want to create the matrix representation for $\alpha$. For that I have to calculate $\alpha(V_i)$ and then write it in the basis $V_i$ to get my values for the matrix. How exactly do I do the last in sage ? For eg. a polynomial ring it's easy because the elements are already written according to its base but in general thats not the case...

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As you said for polynomials, it depends on the way your elements are represented. Could you please provide an example for $\alpha$ and $V_i$ (the one you are dealing with, with the code to produce them) ?

p=5
F.<c>=GF(p^p,modulus=x^p-x-1)
V=[c^i for i in [0..p-1]]
s=c^6+c^3+c^2+1

Now I want to get the coeffecients of s represented to the base V[i], how do i do that

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The following dialog with the sage interpreter (linux console) gives some ideas to proceed:

sage: p = 5
sage: q = p^p
sage: f = GF(p)
sage: R.<x> = f[]
sage: F.<c> = GF( q, modulus=x^p-x-1 )
sage: s = c^6+c^3+c^2+1
sage: s
c^3 + 2*c^2 + c + 1
sage: pol = R(s)    # the lift of s in R
sage: pol
x^3 + 2*x^2 + x + 1
sage: R( c^2+3 ).coefficients() # trap
[3, 1]
sage: R( c^2+3 ).coefficients( sparse=0 )
[3, 0, 1]


We can lift to $\mathbb Z$. (No sense, criptic. Apply then f.)

sage: ZZ( s.integer_representation() ).digits(base=p)
[1, 1, 2, 1]


Of course, the list is to short. So:

sage: L = R(s).coefficients( sparse=0 )
sage: L
[1, 1, 2, 1]
sage: L += (p-len(L)) * [ f(0), ]
sage: L
[1, 1, 2, 1, 0]


With the above notations:

sage: def coeffs( s ):
L = R(s).coefficients( sparse=False )
L += (p-len(L)) * [ f(0), ]
return L
....:
sage: M = matrix( f, p, p, [ coeffs( s*c^k ) for k in range(p) ] )
sage: M
[1 1 2 1 0]
[0 1 1 2 1]
[1 1 1 1 2]
[2 3 1 1 1]
[1 3 3 1 1]


(You want M or M.transpose() .)