I'm trying to solve the following exercise:
Consider the vectorspace $V$ of polynomials of the form $ax + b$, $a, b \in \mathbb{R}$. Find two linear transformations $f_1, f_2 : V \rightarrow V$ for which $f_1 \circ f_2 = \operatorname{id}$ and $f_2 \circ f_1 \neq \operatorname{id}$.
Surely this must be impossible, right? Let $B = \{1, x\}$ be a basis of $V$ and thus:
$$ M_B^B(f_1 \circ f_2) = M_B^B(f_1) M_B^B(f_2) = I $$
such that:
$$ M_B^B(f_2) = M_B^B(f_1)^{-1} $$
and thus:
$$ M_B^B(f_2 \circ f_1) = M_B^B(f_1)^{-1} M_B^B(f_1) = I \implies f_2 \circ f_1 = \operatorname{id} $$
Where am I wrong here?
