MST10030 notes wiki

If $a=\begin{bmatrix}a_1&a_2&\dots&a_n\end{bmatrix}$ is a $1\times n$ row vector and $b=\begin{bmatrix}b_1\\b_2\\\vdots\\b_n\end{bmatrix}$ is an $n\times 1$ column vector, then the row-column product, or simply the product of $a$ and $b$ is defined to be \[ ab=\begin{bmatrix}a_1&a_2&\dots&a_n\end{bmatrix}\begin{bmatrix}b_1\\b_2\\\vdots\\b_n\end{bmatrix}=a_1b_1+a_2b_2+\dots+a_nb_n.\]

If we want to emphasize that we are multiplying in this way, we sometimes write $a\cdot b$ instead of $ab$.

The product $ab$ of a $1\times m$ row vector $a$ with an $n\times 1$ column vector $b$ is undefined if $m\ne n$.