Subspace Dimension

The dimension of a given subspace $U$, $dim(U)$ is the size of its Basis. This will always remain consistent by Invariance Theorem.

Dimension and Spanability

If you have a subspace $S$ of $m$ vectors and $k=dim(U)$

• If $m>k$, $S$ is linearly dependent
• If $m<k$, then $S$ does not span $U$

Concepts

• The Dimension Bound Theorem