Dot Product Associates with Scalar Multiplication/Proof 1
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Theorem
- $\paren {c \mathbf u} \cdot \mathbf v = c \paren {\mathbf u \cdot \mathbf v}$
Proof
| \(\ds \left({c \mathbf u}\right) \cdot \mathbf v\) | \(=\) | \(\ds \sum_{i \mathop = 1}^n \left({c u_i}\right) v_i\) | Definition of Dot Product | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 1}^n c \left({ u_i v_i }\right)\) | Real Multiplication is Associative | |||||||||||
| \(\ds \) | \(=\) | \(\ds c \sum_{i \mathop = 1}^n u_i v_i\) | Real Multiplication Distributes over Real Addition | |||||||||||
| \(\ds \) | \(=\) | \(\ds c \left({\mathbf u \cdot \mathbf v}\right)\) | Definition of Dot Product |
$\blacksquare$