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Definition of vector-scalar multiplication

Created over 8 years ago, updated 3 days ago

Definition: The scalar multiplication of a vector (\mathbf{v} = (v_1, v_2, \ldots, v_n) \in \mathbb{F}^n) by a scalar (a \in \mathbb{F}) is the vector obtained by multiplying each component by (a):

[a \cdot \mathbf{v} = (a \cdot v_1,; a \cdot v_2,; \ldots,; a \cdot v_n)]

Component-wise formula: ((a\mathbf{v})_j = a \cdot v_j) for each (j = 1, 2, \ldots, n).

Examples:

  • In (\mathbb{R}^3): (2 \cdot (1, 3, -1) = (2, 6, -2))
  • In (\mathbb{C}^2): (i \cdot (1+i, 2) = (i-1, 2i))
  • With negative scalar: (-1 \cdot (3, -2, 5) = (-3, 2, -5))

Geometric interpretation ((\mathbb{R}^n)): Scalar multiplication scales the vector by (|a|) and, if (a < 0), reverses its direction.

Algebraic properties: Scalar multiplication satisfies distributivity over vector addition and scalar addition, associativity, and has a scalar identity: (1 \cdot \mathbf{v} = \mathbf{v}).