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Scalar multiplication

a.k.a.

• Vector scaling

Define

… in terms of multiplication as the binary operation

\[\begin{equation*}
  a \vec{v}
  = a \begin{pmatrix} v_1 \\ \vdots \\ v_n \end{pmatrix}
  = \begin{pmatrix} a v_1 \\ \vdots \\ a v_2 \end{pmatrix}
  \in V^n,
\end{equation*}
\]

with

\[\begin{equation*}
  a \in F, \quad
  \vec{v} \in V^n, \quad
  \forall i : v_i \in F,
\end{equation*}
\]

where	\(a\)	is the scalar and
	\(a \vec{v}\)	is the scaled vector,
	\(V^n\)	is the vector space with \(\vec{v}\), and
	\(F\)	is the field over which \(V^n\) is defined.

Imagine

… drawing \(a \vec{v}\) as a copy of \(\vec{v}\) with

• the length \(|a|\) times larger than that of \(\vec{v}\) and
• the direction reversed if \(a < 0\).

Constrain

• Multiplicative identity

  … is \(1\), so

  \[\begin{gather*}
    \forall \vec{v} \in V, \\[1ex]
    1 \vec{v} = \vec{v}.
  \end{gather*}
  \]

  More generally,

  \[\begin{gather*}
    \forall v \in V, \> \exists! a \in F : \\[1ex]
    a \vec{v} = \vec{v},
  \end{gather*}
  \]

  where the identity is

  \[\begin{equation*}
    a = 1.
  \end{equation*}
  \]

• Associative property

  … holds, so

  \[\begin{gather*}
    \forall a, b \in F, \> \forall \vec{v} \in V, \> \\[1ex]
    (a b) \vec{v} = a (b \vec{v}).
  \end{gather*}
  \]

• Distributive property

  … holds over vector addition, where

  \[\begin{gather*}
    \forall a, b \in F, \> \forall \vec{u}, \vec{v} \in V, \> \\[1ex]
    a(\vec{u} + \vec{v}) = a \vec{u} + a \vec{v}
  \end{gather*}
  \]

  which is the one and only

  link between vector addition and scalar multiplication
  

  in the vector space.

Implement

… in Scheme as

• my-vector-scale

  (define (my-vector-scale vector scalar)
    (map (lambda (component)
           (* scalar
              component))
         vector))