The product of a [[Matrix]] and a [[../Math/Vector|Vector]] is a non [[Commutative]] [[Binary Operation]] that produces a [[../Math/Vector|Vector]], and is defined as: $A\vec x$ is the [[../Math/Linear Combination|Linear Combination]] of all column [[Vector|Vectors]] of $A$ $\huge \forall A \in M_{n\times m} ,\vec x \in \R^{m} \ba{A\vec x \in \Rn n} $ $\huge \begin{align} A\vec x &= x_{1}\vec A_{1} + x_{2}\vec A_{2} + \dots + x_{m} \vec A_{n}\\ A\vec x &= \sum_{i=1}^{m} \vec A^{\intercal}_{m} \cdot x_{i} \end{align}$