Let $L$ be a linear transformation from $P_4(\Bbb R)$ to $M(1\times2)(\Bbb R)$ such that $L(1) = [1,\ 1]$, $L(x) = [1,\ 2]$, and $L(x^2) = L(x^3) = L(x^4) = [0,\ 0]$. Do $4$ linearly independent vectors in $\ker(L)$ exist? If so, write them down. Otherwise, prove why it isn't possible.
This question has been confusing me for a while now. Obviously three lin. ind. vectors exist as the question shows, but there doesn't seem to be anything else that would also map to $[0,\ 0]$. Any insight would be appreciated.
The map $L$ is a linear map from $P_4(\Bbb R)$, whose dimension os $5$, into $\Bbb R^{1\times2}$, whose dimension is $2$. The range of $L$ contains two linealry independent vectores ($[1\ \ 1]$ and $[1\ \ 2]$) and so the rank of $L$ is $2$. So, by the rank-nullity theorem, $\dim\ker L=5-2=3$. Therefore, the answer is negative.