Originally Posted by

**Runty** Upcoming midterms are giving me very little time to work on assignments such as this one, so I need help with this.

Suppose $\displaystyle V$ is a vector space over a field $\displaystyle F$.

Suppose $\displaystyle B=\{v_1,...,v_n\}$ is a basis for $\displaystyle V$, $\displaystyle T:V\rightarrow V$ is a linear operator with matrix $\displaystyle A\in F^{n\times n}$ with respect to $\displaystyle B$, $\displaystyle U:V\rightarrow V$ is a linear operator with matrix $\displaystyle D\in F^{n\times n}$ with respect to $\displaystyle B$.

**a)** Prove that the composition $\displaystyle TU:V\rightarrow V$ is a linear. (typo?)

**b)** Prove that the matrix of $\displaystyle TU$ with respect to $\displaystyle B$ is $\displaystyle AD\in F^{n\times n}$ by proving that the $\displaystyle ji$th entry of the matrix of $\displaystyle TU$ is $\displaystyle \sum_{k=1}^n a_{jk}d_{ki}$.

A hint provided is as follows: Expand $\displaystyle U(v_i)$ then use this expansion to expand $\displaystyle T(U(v_i))$.

This is probably quite easy, but as I said before, I can't devote too much time to this or I'll not be able to properly prepare for my midterms.