I'm reading through Axler Sheldons Linear Algebra Done Right, and stumbled uppon the following question:
Suppose $V$ is an inner-product space and $v,w \in \mathcal{L} (V)$. Define $T \in \mathcal{L} (V)$ by $Tu = \langle u,v \rangle w$. Find a formula for $trace (T)$.
The thing I can't wrap my head around is that $v$ is transformation (or matrix), and we're trying to take the inner product between a matrix and a vector.
If I assume that the $\mathcal{L}$ is nothing but a typo, then the whole statement makes sense, but I'm just wondering:
Is the inner product between a vector and transform a valid operation?