Example: Trivial Subspaces, Zero Space

(related to Definition: Subspace)

Please note that every vector space \(V\) over a given field \(F\) has two trivial subspaces:

  1. \(V\) itself is a subspace of \(V\), and
  2. The subset \(U:=\{0_V\}\) consisting of the zero vector of \(V\) is also a subspace of \(V\).

\(U\) is sometimes also called the zero space. The vector \( 0_V \) has to exist, since \((V,\oplus)\) is a group, which is part of the definition of vector space. The index of the vector \(0_V\) is used to differentiate it from the zero element of the addition \(0\in F\), and can be omitted, if it is clear from the context, which zero element is meant.

Thank you to the contributors under CC BY-SA 4.0!