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Subspace

A subset the real space \(H \in \mathbb{R}^n\) that

• contains the zero vector \(\vec{0}\), so

  \[\begin{equation*}
    \vec{0} \in H,
  \end{equation*}
  \]

• is closed under vector addition, so

  \[\begin{equation*}
    \forall \vec{u} \in H, \; \forall \vec{v} \in H
    \iff
    (\vec{u} + \vec{v}) \in H, \enspace \qand*
  \end{equation*}
  \]

• is closed under scalar multiplication, so

  \[\begin{equation*}
    \forall c \in \mathbb{R}^1, \; \forall \vec{v} \in H
    \iff
    c \vec{v} \in H.
  \end{equation*}
  \]

• Synonyms
• Examples