1.1. What affects entries in a product

From § you can see this:

Proposition. In a matrix product $AB$,

(i) row $i$ of $AB$ is affected only by row $i$ of $A$; in fact, it's that row times $B$.

(ii) column $j$ of $AB$ is affected only by column $j$ of $B$; in fact, it's $A$ times that column.

Problem V-2. (a) For $C = AB$, fill in all entries of $C$ that you know if $A = \left[\begin{array}{ccc}\cdot& \cdot& \cdot\ 2& 3& 5\ \cdot& \cdot& \cdot\end{array}\right]$ and $B = \left[\begin{array}{rrr}0&1&0\ 10&10&10\ 1&0&0\end{array}\right]$.

(b) The middle row of $C$ is a linear combination of the rows of $B$. With what coefficients?

Problem V-3. If you need to transform 1000 points of $\mathbb{R}^2$ by $\tau _ M$ for some $2 \times 2$ matrix $M$, instead of doing 100 separate matrix multiplications, you can make them the columns of a matrix $P$ and find $MP$. Explain why that works.