I have this question which has two polynomials $$f(t) = t^{3} − 4t^{2} + 3t + 3$$ $$g(t)= t^{3} +2t^{2} +4t -1$$ $$h(t)= 2t^{3} − t^{2} − 3t + 5$$ and I was asked to find whether they are dependent or not. So what I did was, that I took the standard basis in $\mathbb{P}^{3}[\mathbb{R}] \rightarrow \mathbb{P}^{3}[\mathbb{R}]$ which is $\{1, t, t^{2}, t^{3} \}$. So vector form representation of $f(t)$ and $g(t)$ is $$f = [3,3,-4,1]$$ $$g = [-1, 4, 2, 1]$$ $$h=[5,-3,-1,2]$$ If I write this $af+bg+ch=0 \textrm{ } \forall \textrm{ } a, b, c \textrm{ } \in \textrm{ } \mathbb{R}$, then only possible value for $a, b$ and $c$ is 0, isn't it? So they are indeed independent. The second part was to find the derivative matrix in this given basis. The matrix which I calculated is $$\begin{bmatrix} 0&1&0&0\\0&0&2&0\\0&0&0&3\\0&0&0&0\end{bmatrix}$$. I know that this matrix is correct if the basis is $\{1, t, t^{2}, t^{3} \}$. But I am not sure what basis the question is talking about. Question
So I am not sure whether I posted the question correctly in mathematical language. If you have any doubt in the way I presented the question, please have look at the image.