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Definition of scalar

Created over 8 years ago, updated 3 days ago

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})