Subspaces

subspaces

Definition

Given a vector space \(V\), a subset \(U\) of \(V\) is a subspace if it is a vector space with respect to the addition and scalar multiplication operations inherited from \(V\).

Examples

  • \(\{0\}\) and \(V\) are subspaces of every vector space \(V\) and are therefore trivial examples.
  • Non-trivial subspaces of \(\mathbb{R}^{2}\) include all lines passing through the origin.
  • Non-trivial subspaces of \(\mathbb{R}^{3}\) include all lines and planes passing through the origin.

Algorithm

To determine if a subset \(U\) of \(V\) is a vector space, perform these three checks:

  • Check if \(0 \in U\). Every vector space should have at least the zero element.
  • For arbitrary \(u, v \in U\) and \(a \in \mathbb{R}\), check if:
    • \(u + v \in U\)
    • \(av \in U\)
  • If all three checks are successful, then \(U\) is a subspace of \(V\). \(U\) is not a subspace of \(V\) even if one of these three checks fails.

For example, let \(U = \{x + y + z = 0\ |\ x, y, z \in \mathbb{R}^3\} \subset V\). \(U\) is a subspace of \(\mathbb{R}^{3}\) as:

  • \((0, 0, 0) \in U\)
  • Let \((x_1, y_1, z_1) \in U\) and \((x_2, y_2, z_2) \in U\), then we know that \(x_1 + y_1 + z_1 = x_2 + y_2 + z_2 = 0\). From this, we can infer:
    • \((x_1 + x_2) + (y_1 + y_2) + (z_1 + z_2) = 0\) which implies \((x_1, y_1, z_1) + (x_2, y_2, z_2) \in U\)
    • \(ax_1 + ay_1 + az_1 = 0\) which implies \(a(x_1, y_1, z_1) \in U\)