% m7_expansions.m - coefficients and expansions

A = [1 2 3; 4 5 6; 2 2 0]
A*[1;0;1]
A*[1;-2;0]
inv(A)*[2;2;2]
A\[2;2;2]

t = (0:.1:2)' 
plot(t,exp(-t),'r'), grid on

A = [t.^0 t.^1 t.^2]           % this is called a Vandermonde matrix  
x = [1; -1; 1/2]
hold on, plot(t,A*x,'--b')

x = A\exp(-t)   % backslash for rectangular A: least-squares approx. 
plot(t,A*x,'--k')
legend('exp(-t)','Taylor','least-squares','location','southwest')

hold off, plot(t,A*x-exp(-t)), grid on