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

$\blacksquare$