Matrix-Vector Multiplication

How do we multiply matrix $A$ by $x$?

$$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=?$$

Method 1: Coefficients of a Linear Combination

Think about the entries of $x$ as coefficients for a linear combination of the column vectors of $A$, such that:

$$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=1\left[\begin{array}{c}2\\ 1\end{array}\right]+2\left[\begin{array}{c}5\\ 3\end{array}\right]=\left[\begin{array}{c}12\\ 7\end{array}\right]$$

Method 2: Dot Product of Rows

Take the dot product of each row of $A$ with a vector $x$:

$$\left[\begin{array}{cc}2& 5\\ 1& 3\end{array}\right]\left[\begin{array}{c}1\\ 2\end{array}\right]=\left[\begin{array}{c}2\cdot 1+5\cdot 2\\ 1\cdot 1+3\cdot 2\end{array}\right]=\left[\begin{array}{c}12\\ 7\end{array}\right]$$