So the question asks that:

Let $b_1=\begin{bmatrix}-1\\-2\end{bmatrix}$and $b_2=\begin{bmatrix}-2\\-5\end{bmatrix}$The set 𝔅={−1−2t,−2−5t} is a basis for $P_1$. Suppose that $T:P_1→P_1$ is a linear transformation whose 𝔅-matrix, $B$, is $B=$\begin{bmatrix}1&2\\-2&0\end{bmatrix} Find the matrix $A$ of $T$ relative to the standard basis {1,t} for $P_1$.

So I know in order to find a matrix relative to a basis, I normally do $T(x_1), T(x_2)$ where $x_1$ and $x_2$ are the vectors or the relative basis. So in order to find A, I should do $T(1)$ and $T(t)$ and express them in {1,t}.

However, since I am not given the linear transformation formula, how do I suppose to find A?

Or should I do A by treating A as a similar matrix of B?


Hint: Let $b_1=-1-2t$ and $b_2=-2-5t$ and express both elements of the requested basis $(1,t)$ by linear combinations of $b_1$ and $b_2$, then use linearity of $T$.


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