linear independent set formation

show that {1,cost,cos2t.......cos6t} is linearly independent

using matlab code

can anyone help

Answer 1

[Bruno Luong]

t = rand(1,7)*2*pi;
independent = rank(cos((0:6)'.*t)) == 7

independent = logical
   1

[sanket neharkar]

actually its not 2t,3t...6t

its the power

[sanket neharkar]

like cos (square)t, cos (cube)t.......cos^6t

[Matt J]

@sanket neharkar but surely you can modify the answer appropriately.

[Bruno Luong]

t = rand(1,7)*2*pi;
independent = rank(cos(t).^((0:6)')) == 7

independent = logical
   1

Answer 2

[Paul]

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

syms t
for ii = 0:6
    eqn(ii+1) = diff(sum(a.*cos(t).^(0:6)),t,ii);
end

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

simplify(det(M))

ans = 

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.