Vector Space Dimension

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

Dimension and Span-ability

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
• Subspace Finite Dimensional Vector Space Lemma
• Dimension and Injectivity Theorem
• Dimension and Surjectivity Theorem