Given a vector space $V =\{a\cos(t) + b\sin(t) + ct\sin(t)\}$ with $a,b,c$ all real, a basis $B = \{cos(t),sin(t),t\sin(t)\}$ and a Linear operator $Lf(t) = f''(t)$, how would I write ${\lbrack L\rbrack}_{BB}$?
I attempted it and got \begin{bmatrix} -1 & 0 & 2t \\ 0 & -1 & 0 \\ 0 & 0 & -t \end{bmatrix}
however my professor shows the matrix representation as being
\begin{bmatrix} -1 & 0 & 2 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{bmatrix}
and so I am unsure as to why the $t$ disappeared