Matrix Scalar Product Distributes over Number Addition

Theorem

Let $\Bbb F$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.

For $\mathbf A \in \map \MM {m, n}$ and $\lambda$ \in $\Bbb F$, let $\lambda \mathbf A$ be defined as the matrix scalar product of $\lambda$ and $\mathbf A$.

The matrix scalar product is associative on $\map \MM {m, n}$, in the following sense:

For all $\mathbf A$ in $\map \MM {m, n}$ and $\lambda, \mu \in \Bbb F$:

$\paren {\lambda + \mu} \mathbf A = \lambda \mathbf A + \mu \mathbf A$

Proof

 $\ds \paren {\lambda + \mu} \mathbf A$ $=$ $\ds \paren {\lambda + \mu} \sqbrk a_{m n}$ Definition of $\mathbf A$ $\ds$ $=$ $\ds \sqbrk {\paren {\lambda + \mu} a}_{m n}$ Definition of Matrix Scalar Product $\ds$ $=$ $\ds \sqbrk {\lambda a + \mu a}_{m n}$ Multiplication of Numbers Distributes over Addition $\ds$ $=$ $\ds \sqbrk {\lambda a}_{m n} + \sqbrk {\mu a}_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \lambda \mathbf A + \mu \mathbf A$ Definition of $\mathbf A$

$\blacksquare$