11.3 Matrix–vector multiplication

In Module 11.2, a matrix product connected two grids. What if we have just one set of input values? The vector from Module 10.3 is a one-column grid, so the same multiplication rule turns it into one set of outputs. Each row states how to combine the inputs for one output, whether that output is a score or a new coordinate.

This is intermediate linear algebra. Learn matrix products in OpenStax, then treat the vector as a one-column matrix.

A basic idea: one row, one output

If $A$ has shape $m\times n$ and $x$ is an $n\times1$ vector, then $Ax$ is an $m\times1$ vector. Row $i$ of $A$ produces output $y_i=\sum_{j=1}^{n}a_{ij}x_j$.

For example,

$$\begin{bmatrix}2&1\\-1&3\end{bmatrix} \begin{bmatrix}4\\2\end{bmatrix} =\begin{bmatrix}2(4)+1(2)\\-1(4)+3(2)\end{bmatrix} =\begin{bmatrix}10\\2\end{bmatrix}.$$

A quick tour of properties

The same result is a combination of columns: if $A=[a_1\ \cdots\ a_n]$, then $Ax=x_1a_1+\cdots+x_na_n$. Here each input $x_j$ scales one whole column, and the scaled columns add to the output. This is the same arithmetic seen from the input side rather than one output row at a time. Dimensions tell you whether the product is defined: a vector with the wrong number of entries cannot be multiplied by $A$. Matrix–vector multiplication is linear: $A(u+v)=Au+Av$ and $A(cu)=cAu$. Combining inputs before applying $A$ gives the same result as combining their outputs afterward.

Application 1: weighted scores

Two candidates have feature vectors $x=(3,4)$ and $z=(5,1)$. Score one uses weights $(2,1)$ and score two uses $(1,3)$. Stacking the weights as rows gives

$$A=\begin{bmatrix}2&1\\1&3\end{bmatrix}.$$

Then $Ax=(10,15)$ and $Az=(11,8)$. Candidate $z$ leads on the first score; $x$ leads on the second. Each row states how that score weights the features.

Application 2: a coordinate transformation

Let $T=\begin{bmatrix}0&1\\1&0\end{bmatrix}$. For $v=(2,3)$,

$$Tv=\begin{bmatrix}0\cdot2+1\cdot3\\1\cdot2+0\cdot3\end{bmatrix}=\begin{bmatrix}3\\2\end{bmatrix}.$$

The matrix swaps the coordinates: it reflects every point in the line $y=x$. This is a geometric transformation rather than a score, but it uses the same row-by-vector arithmetic.