Two ways to think about matrix-vector multiplication
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Useful post about how one can think about matrix-vector multiplication in two different ways, which can be useful in different contexts.
Lets take an example of a matrix-vector multiplication: \(\textbf{A}\vec{b} = \vec{c}\).
Assume the dimensions of \(\textbf{A}\) to be \((3,2)\), \(\vec{b}\) to be \((2,1)\), \(\vec{c}\) to be \((3,1)\). Hence, we have:
The more common way to think of the resultant vector is as a collection of the dot products of each row of \(\textbf{A}\) with the only column in \(\vec{b}\):
\[\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} [p,q] \cdot [v,w] \\ [r,s] \cdot [v,w] \\ [t,u] \cdot [v,w] \end{pmatrix} = \begin{pmatrix} p \cdot v + q \cdot w \\ r \cdot v + s \cdot w \\ t \cdot v + u \cdot w \end{pmatrix}\]The other way is to think of the resultant vector as a sum of two vectors, which are a weighted sum of the columns of \(\textbf{A}\), according to the values in \(\vec{b}\).
\[\begin{pmatrix} x \\ y \\ z \end{pmatrix} = v \cdot \begin{pmatrix} p \\ r \\ t \end{pmatrix} + w \cdot \begin{pmatrix} q \\ s \\ u \end{pmatrix} = \begin{pmatrix} p \cdot v + q \cdot w \\ r \cdot v + s \cdot w \\ t \cdot v + u \cdot w \end{pmatrix}\]