# Scalar Product with Sum

## Theorem

Let $\struct {G, +_G}$ be an abelian group whose identity is $e$.

Let $\struct {R, +_R, \times_R}$ be a ring whose zero is $0_R$.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

Let $x \in G, \lambda \in R$.

Let $\sequence {x_m}$ be a sequence of elements of $G$.

Then:

$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^m x_k} = \sum_{k \mathop = 1}^m \paren {\lambda \circ x_k}$

## Proof

This follows by induction from Module Axiom $\text M 1$: Distributivity over Module Addition, as follows:

For all $m \in \N_{>0}$, let $\map P m$ be the proposition:

$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^m x_k} = \sum_{k \mathop = 1}^m \paren {\lambda \circ x_k}$

### Basis for the Induction

$\map P 1$ is true, as this just says:

$\lambda \circ x_1 = \lambda \circ x_1$

This is our basis for the induction.

### Induction Hypothesis

Now we need to show that, if $\map P n$ is true, where $n \ge 1$, then it logically follows that $\map P {n + 1}$ is true.

So this is our induction hypothesis:

$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^n x_k} = \sum_{k \mathop = 1}^n \paren {\lambda \circ x_k}$

Then we need to show:

$\ds \lambda \circ \paren {\sum_{k \mathop = 1}^{n + 1} x_k} = \sum_{k \mathop = 1}^{n + 1} \paren {\lambda \circ x_k}$

### Induction Step

This is our induction step:

 $\ds \lambda \circ \paren {\sum_{k \mathop = 1}^{n + 1} x_k}$ $=$ $\ds \lambda \circ \paren {\sum_{k \mathop = 1}^n x_k + x_{n + 1} }$ $\ds$ $=$ $\ds \lambda \circ \paren {\sum_{k \mathop = 1}^n x_k} + \lambda \circ x_{n + 1}$ Module Axiom $\text M 1$: Distributivity over Module Addition $\ds$ $=$ $\ds \sum_{k \mathop = 1}^n \paren {\lambda \circ x_k} + \lambda \circ x_{n + 1}$ Induction hypothesis $\ds$ $=$ $\ds \sum_{k \mathop = 1}^{n + 1} \paren {\lambda \circ x_k}$

So $\map P n \implies \map P {n + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:

$\ds \forall m \in \N_{>0}: \lambda \circ \paren {\sum_{k \mathop = 1}^m x_k} = \sum_{k \mathop = 1}^m \paren {\lambda \circ x_k}$

$\blacksquare$