1.2. Subspaces#
Let \(U\) be a subset of the vector space \(V\). We say that \(U\) is a vector subspace of \(V\)(or simply subspace of \(V\)) if \(U\) is also a vector space with the same addition and scalar multiplication defined on \(V\).
To check whether a given subset \(U \subseteq V\), we may simply check that if \(U\) contains the zero vector and if it is closed under the addition and scalar multiplication.
Proposition 1.7
Let \(U\) be a subset of \(V\). Then \(U\) is a vector subspace of \(V\) if and only if
\(\mathbf{0} \in U\),
\(\mathbf{u} + \mathbf{v} \in U\) for all \(\mathbf{u}, \mathbf{v} \in U\), and
\(a \mathbf{u} \in U\) for all \(a \in \F\) and \(\mathbf{u} \in U\).
By simple observations, one may notice that the additive identity in the subspace \(U\) is exactly the one in the superspace \(V\) and \(\mathbf{w}\) is the additive inverse of \(\mathbf{u}\) in \(U\) if and only if it is also the additive inverse of \(u\) in \(V\).
Example 1.1
\(\{\mathbf{0}\}\) and \(V\) are subspaces of \(V\), which are the simplest examples of vector spaces.
Example 1.2
\(\{(x, 0, 0) \mid x \in \F\}\) and \(\{(x, y, 0) \mid x, y \in \F\}\) are subspaces of \(\F^3\). Specially, when \(\F = \R\), this means that the 1D \(x\)-axis line and the 2D \(x\)-\(y\) plane are subspaces of the 3D space.
In fact, all 1D lines and 2D planes that pass through the origin are subspaces of the 3D space.