Matrix-Vector Products
Definition
The matrix-vector product between an $m \times n$-matrix and an $n$-vector is the result of the linear combination of the columns of the matrix with the sequence of scalars in the vector. So the matrix-vector product is an $m$-vector.
Notation
Let $C$ be a nonempty set.
Let $A \in \Mat{C}{m}{n}$ and let $x \in
\Vec{C}{n}$.
As usual, we denote the matrix-vector product
of $A$ with $x$ by $Ax$, read “A x,” and
defined by
\[
b_i = \sum_{j = 1}^{n} a_{ij}x_j
\] \[
A = \bmat{
& a_1^\tp & \\
& a_2^\tp & \\
& \vdots & \\
& a_{m}^\tp &
}
\] \[
b_i = a_i^\top x
\]
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