Dot Product Operator is Bilinear

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Theorem

Let $\mathbf u, \mathbf v, \mathbf w$ be vectors in the real Euclidean space $\R^n$.

Let $c$ be a real scalar.

Then:

$\paren {c \mathbf u + \mathbf v} \cdot \mathbf w = c \paren {\mathbf u \cdot \mathbf w} + \paren {\mathbf v \cdot \mathbf w}$

Proof

$\ds \paren {c \mathbf u + \mathbf v} \cdot \mathbf w$

$=$

$\ds c \sum_{i \mathop = 1}^n \paren {u_i + v_i} w_i$

Definition of Dot Product

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n \paren {c u_i + v_i} w_i$

Real Multiplication Distributes over Real Addition

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n \paren {c u_i w_i + v_i w_i}$

Real Multiplication Distributes over Real Addition

$\ds $

$=$

$\ds \sum_{i \mathop = 1}^n c u_i w_i + \sum_{i \mathop = 1}^n v_i w_i$

Real Multiplication is Commutative

$\ds $

$=$

$\ds c \sum_{i \mathop = 1}^n u_i w_i + \sum_{i \mathop = 1}^n v_i w_i$

Real Multiplication Distributes over Real Addition

$\ds $

$=$

$\ds c \paren {\mathbf u \cdot \mathbf w} + \mathbf v \cdot \mathbf w$

Definition of Dot Product

$\blacksquare$