Dimension

The dimension of a Subspace $H$ is the Cardinality of its Basis $\mathcal{B}$

$$\dim H=|\mathcal{B}|$$

$\dim \{\vec{0}\}$

$$\dim \vec{0}=0$$

This is because a basis requires a Set of Linearly Indepedent Vectors, and $\vec{0}$ does not meet this requirement. Therefore, its only basis is $\varnothing$. The cardinality of the empty set is 0, therefore so is $\dim \vec{0}$

Examples

• dim ${\mathbb{R}}^n$
  • $n$
• dim(Nul A) is the number of
  • number of free vars of A
• dim(Col A) is the number of
  • number of pivots of A
• H = $\{(x_1....,x_n):x_1+...+x_n=0\}$
  • n variables, solve for $x_1$ ito everything else. → one pivot everything else free vars. Therefore $n-1$ free vars