The question is given below:
My first guess is:
1-for $t = (2n + 1)\pi$, where $n \in \mathbb{Z}$. we always have $\cos(t) = -1$ and $\sin(t) = 0$ so we will get the vectors $(-1, 1)^{t}$ and $(1, -1)^{t}$ which are linearly dependent.
2- for $ t = 2n\pi$, where $n \in \mathbb{Z}$. the two vectors will become the same vector $(1, 1)^t$ and hence are linearly dependent.
But what about the rationals and irrationals and the other values of $t$ in the integers ?
So I started to row reduce the following matrix:
$$\begin{bmatrix} \cos(t) + i\sin(t)&1\\ 1&\cos(t) - i\sin(t)\\ \\ \end{bmatrix}$$
But I am stucked now, could anyone help me please?
Thanks!