Proving a set is linearly independent - MATLAB

Question

I am trying to show that the set $\{1, \cos(t), \cos^2(t), \dots, \cos^6(t)\}$ is a linearly independent set.

I am able to use MATLAB.

However, I am not so sure on how to show that none of these results are a scalar multiple of themselves, or a linear combination of other vectors..

I saw this post here regarding the same problem: Showing that $\{ 1, \cos t, \cos^2 t, \dots, \cos^6 t \}$ is a linearly independent set, however, it did not really help me make progress.

I would appreciate it if someone could help me figure this problem out.

Answer 1

You can try with the Wronskian of the set $S = \{1,\cos t,\cdots,\cos^6t\}$. In general if a set of function $\{f_1, f_2, \cdots, f_n\}$ is linearly dependent in an interval $I$ then $W(x\in I) = 0$, where

$$ W(f_1, f_2,\cdots, f_n)(x) = \left|\begin{array}{cccc} f_1^{(0)}(x) & f_2^{(0)}(x) & \cdots & f_n^{(0)}(x) \\ f_1^{(1)}(x) & f_2^{(1)}(x) & \cdots & f_n^{(1)}(x) \\ && \vdots & \\ f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x) \\ \end{array}\right| $$

where $f_j^{(i)}$ means the $i$-the derivative of the $j$-th function. To calculate this determinant you can use symbolic variables in Matlab, and this is what you should get

$$ W(1,\cos t\cdots, \cos^6t)(t) = -24~883~200 \sin^{21}(t) $$

which should tell you in which interval the set $S$ is linearly independent

Answer 2

I think that in MatLab it would be sufficient to draw a random sample of values of $t$ and to then generate a Gramian matrix. You will need seven values of $t$ as there are seven functions. Denote these $t_0, \ldots, t_6$.

Consider the vectors

$$ v_0 = (1,1,1,1,1,1,1)^T $$ $$ v_1 = (\cos t_0, \cos t_1, \ldots, \cos t_6)^T $$ $$ v_2 = (\cos^2 t_0, \cos^2 t_1, \ldots, \cos^2 t_6)^T $$ $$ \vdots $$ $$ v_6 = (\cos^6 t_0, \cos^6 t_1, \ldots, \cos^6 t_6)^T $$

Build the matrix $V$ that contains these vectors as its columns. The Gramian matrix is easily calculated as $V^T V$ and it has a zero determinant if and only if the vectors are linearly dependent.

If the original functions were linearly dependent for all $t$ then the vectors we generate for a random set of values would remain linearly dependent. In other words, it would be sufficient to show that you don't get a zero determinant to show that the functions are not linearly dependent for all $t$.