I have a linear transformation $F: \mathbb R^3 \to \mathbb R[t]_2$, where $\mathbb R[t]_2$ is the vector space of polynomials up to the second degree.
I also have three different bases:
$$B = \{(1,1,0),(0,1,0),(0,0,1)\}$$
$$C = \{1, 1 + t, 1 + t^2\}$$
$$D = \{1, t, t^2\}$$
The linear transformation matrix $[F]_{B,C}$ has also been given:
$$\begin{pmatrix}1&2&-1\\1&0&-1\\0&1&0\end{pmatrix}.$$
The problem asks to write the linear transformation matrix of $[F]_{B,D}$.
What I've tried so far:
$$F(1,1,0) = something = 1(1,0,0) + 1(1,1,0) + 0(1,0,1)$$
$$F(0,1,0) = something = 2(1,0,0) + 0(1,1,0) + 1(1,0,1)$$
$$F(0,0,1) = something = (-1)(1,0,0) + (-1)(1,1,0) + 0(1,0,1)$$
So this is basically my attempt to find the linear transformation formula so I can write the matrix. But I'm not sure what to put where I wrote "something" as I'm dealing with a polynomial.
I've also thought that there may be a way to find the $[F]_{B,D}$ matrix using matrices of changing basis, but I could not think how.
Can anyone give me any hint?
Best Answer
If you denote the identity by $I:R[t]_2\rightarrow R[t]_2$, the corresponding representation from basis $C$ to basis $D$ is (just reading from the given polynomials)
$$[I]_{C,D} =\begin{pmatrix}1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}$$
So, the matrix looked for is
$$[F]_{B,D} = [I]_{C,D}[F]_{B,C} = \begin{pmatrix}1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\begin{pmatrix}1&2&-1\\1&0&-1\\0&1&0\end{pmatrix}=\begin{pmatrix}2&3&-2\\1&0&-1\\0&1&0\end{pmatrix}$$
Tudo claro agora?