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*}
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