linear independent set formation
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Paul on 5 Sep 2022
A set of functions fi(t) is linearly indepenedent if the only solution to
a1*f1(t) + a2*f2(t) .... an*fn(t) = 0 (1)
is ai = 0 for all values of t.
If eqn(1) is true, then derivatives of the lhs are also zero.
Define the coefficients ai
a = sym('a',[1 7]);
Define the test eqn and its derivatives
for ii = 0:6
eqn(ii+1) = diff(sum(a.*cos(t).^(0:6)),t,ii);
Put the set of equations into the form M*a = 0
[M,b] = equationsToMatrix(eqn,a); % b = 0
The only way this equation has a nonzero solution for a for all values of t is if M is singular for all t. Check the determinant
It is, in general, nonzero, so the only solution for all t that satisfies (1) is ai = 0. Therefore, the functions cos(t)^n (n=0:6) are linearly independent.