I have this homework question that I have no idea how to do:
Show that $\{t, \sin(t), \cos(2t), \sin(t)\cos(t) \}$ is a linearly independent set of functions defined on $\mathbb{R}$. Start by assuming that $$c_1 t + c_2 \sin(t) + c_3 \cos(2t) + c_4 \sin(t)\cos(t) = 0$$ for all $t$. Choose specific values of $t$ ($t=0, 1, 2\dots$) until you get a system with enough equations to determine that all the $c_i$'s must be $0$.
My only guess is to set up a matrix based on this polynomial somehow, then row reduce it to find the pivot columns. Subbing in different $t$'s would allow for some value to be attached to each $c$. Would that be the matrix? Shouldn't it be formed by the $c$ values?