Definition of scalar

**Definition:** A *scalar* is a single number from a field \(\mathbb{F}\), typically \(\mathbb{R}\) (real numbers) or \(\mathbb{C}\) (complex numbers).

Scalars are used to multiply vectors in scalar multiplication. The field \(\mathbb{F}\) determines what kinds of scalars are allowed:

- When working over \(\mathbb{R}\), scalars are real numbers: \(3\), \(-1.5\), \(\pi\), etc.
- When working over \(\mathbb{C}\), scalars are complex numbers: \(2 + 3i\), \(-i\), \(5\), etc.

**Notation:** Scalars are typically denoted by lowercase italic letters: \(a\), \(b\), \(c\), etc.

**Key property:** Scalars commute with vector operations and distribute over vector addition:
- \(a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v}\)
- \((a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u}\)Copy