Nullspace and Range

• Nullspace (kernel)
  • $N(T)=\{x\in V:T(x)=0\}$
  • All the vectors that get transformed to 0
  • Subspace of the domain
• Range (image)
  • $R(T)=\{T(x):x\in V\}$
  • All the vectors produced by the transform
  • Subspace of the codomain
  • If $\beta =\{v_1,\dots ,v_n\}$ is a basis for the domain $V$, $R(T)=\text{span}(T(v_1),\dots ,T(v_n))$ (theorem 2.2)

Nullity and Rank

• $\text{nullity}(T)=\dim (N(T))$
• $\text{rank}(T)=\dim (R(T))$
• Rank-Nullity/Dimension Theorem: $\text{nullity}(T)+\text{rank}(T)=\dim (V)$

Transformation Insights

• $T$ is one-to-one $⟺N(T)=\{0\}$
• If $\dim (V)=\dim (W)$ finite, the following are equivalent
  • $T$ is one-to-one
  • $T$ is onto
  • $\text{rank}(T)=\dim (V)$