1. (10 points) Recall that $M_{n \times n}(\mathbb{R})$ is the vector space of $n \times n$ matrices with real coefficients. Let A and B be two given $n \times n$ matrices such that B is invertible. Is the set \begin{equation*} \{X \in M_{n \times n}(\mathbb{R}) \mid AXB^{-1} = X\} \end{equation*} a vector space? Justify your answer.
2. (10 points) Recall that $C[0, 1]$ is the vector space of continuous real-valued functions on the interval $[0, 1]$. Is the subset \begin{equation*} \{f(x) \in C[0, 1] \mid \text{derivative } \frac{df}{dx} \text{ exists for all } 0 < x < 1\} \subset C[0, 1] \end{equation*} a vector subspace? Justify your answer.
