###### Definition2.2.2

The product of a matrix $$A$$ by a vector $$\xvec$$ will be the linear combination of the columns of $$A$$ using the components of $$\xvec$$ as weights.

If $$A$$ is an $$m\times n$$ matrix, then $$\xvec$$ must be an $$n$$-dimensional vector, and the product $$A\xvec$$ will be an $$m$$-dimensional vector. If

\begin{equation*} A=\left[\begin{array}{rrrr} \vvec_1 \amp \vvec_2 \amp \ldots \amp \vvec_n \end{array}\right], \xvec = \left[\begin{array}{r} c_1 \\ c_2 \\ \vdots \\ c_n \end{array}\right], \end{equation*}

then

\begin{equation*} A\xvec = c_1\vvec_1 + c_2\vvec_2 + \ldots c_n\vvec_n\text{.} \end{equation*}
in-context