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Definition of size of a vector

Created over 8 years ago, updated 3 days ago

Definition: The size (or dimension) of a vector (\mathbf{v} \in \mathbb{F}^n) is the number (n) of its components.

If (\mathbf{v} = (v_1, v_2, \ldots, v_n)), then the size of (\mathbf{v}) is (n).

Notation: The size of (\mathbf{v}) is often denoted as:

  • (\text{size}(\mathbf{v}) = n)
  • (\mathbf{v} \in \mathbb{F}^n) (the superscript indicates the size)

Examples:

  • (\mathbf{v} = (3, -1, 0, 5)) has size 4, so (\mathbf{v} \in \mathbb{R}^4)
  • (\mathbf{w} = (1+i, 2-i)) has size 2, so (\mathbf{w} \in \mathbb{C}^2)

Important: Two vectors can only be added if they have the same size. Vector operations are defined component-wise, so mismatched sizes make the operation undefined.