u = {0.43897754604943806`,0.598385054145514`,0.9727438713783174`,0.41015894702010325`,0.07276447441095724`,0.895450228637269`,0.5166314722086436`};
p = {{-0.11766288806517933`,-0.9112411814898387`},{0.4962142406313803`,0.9404764517552069`},{-0.19312682412231208`,0.7528692817356553`},{-0.21512291968760122`,-0.5492097324624199`},{0.44926381204412813`,-0.587474226339622`},{-0.025435108077836777`,0.8874183726242024`},{0.5106615291357914`,-0.7659609534980718`}};
c={0.1,0.1};
s=Norm[p[[-1]]-c];
phi=(Exp[-#^2]&);
ps=Table[(q-c)/s,{q,p}];
f[x_,c_,p_,phi_,s_]:=Sum[alpha[i] phi[Norm[x-p[[i]]]],{i,Length@p}]+beta[1] 1+ {beta[3],beta[2]}.x
g[q_]:=f[{q[[1]],q[[2]]},c,ps,phi,s]

ps

g[{x,y}]

interpEq = Table[g[ps[[i]]]==u[[i]],{i,Length[p]}];
ortEq = {Sum[alpha[i] 1,{i,Length[p]}]==0, Sum[alpha[i] ps[[i,2]],{i,Length[p]}]==0,Sum[alpha[i] p[[i,1]],{i,Length[p]}]==0};

vars=Join[Table[alpha[i],{i,Length@p}],{beta[1],beta[2],beta[3]}]
{b,A}=CoefficientArrays[Join[interpEq,ortEq],vars];

Normal[A]//MatrixForm

coef=LinearSolve[A,-b]

{-2.5839802672432306`,0.05949468465926759`,0.3070276203705349`,2.6614384716333603`,-9.109317862041848`,0.06397458900968854`,8.601362763612228`,0.7723570730895989`,-0.17108612429083783`,-0.11758425393124862`}