If \(A\) is a \(m \times n\)-matrix, \(C\) is a \(n \times p\) matrix, and \(B=AC\text{,}\) then \(B\) is the \(m \times p\) matrix given by

\begin{equation*} b_{ij} = \sum_{k=1}^n a_{ik}c_{kj}. \end{equation*}

But in keeping with the ideas above, let us consider the formula for column \(j\) of \(B\text{:}\)

\begin{equation*} b_j = \sum_{k=1}^n c_{kj} {a_k}. \end{equation*}

So column \(j\) of \(B\) is a linear combination of all the columns of \(A\text{,}\) with the coefficients taken from column \(j\) of \(C\text{.}\)