(Range and Null Space of linear transformations)

Range

The range (or image) of a linear transformation $T:V \to W$ is the set of vectors $T$ can output.

$\text{range}(T_M)=\text{col}(M)=\{\vec{y}\in W : \vec{y} = T_M(\vec{x}) \text{ for some } \vec{x} \in V\}$

• If $T:\mathbb{R}^c\to \mathbb{R}^r$ is a linear transformation, then $\text{range}(T)\sub \mathbb{R}^r$ is a subspace.

Range ≠ Codomain

• $\text{range}(T) \sub \text{codomain}(T)$
• Range is the actual outputs whereas codomain is the possible outputs.

Rank

For a linear transformation $T:\mathbb{R}^n \to \mathbb{R}^m$,

The rank of $T$ is the dimension of $\text{range}(T)$

$\text{rank}(T)=\text{dim}(\text{range}(T))$

Note: A plane in $\mathbb{R}^3$ has dimension 2, and the dimension of $\mathbb{R}^3$ is 3!!

Null Space

The null space or kernel of a linear transformation $T: V \to W$ is the set of vectors that get mapped to the zero vector under $T$

$\text{null}(T)=\{ \vec{x} \in V : T(\vec{x}) =\vec{0}\}$

Null-Space Properties

• If $T: ℝ^n \to ℝ^m$ is a linear transformation, then $\text{null}(T) \sub ℝ^n$ is a subspace
• $\text{null}(T_M)=\text{null}(M)$ is the set of all vectors orthogonal to the rows of $M$
• $\text{null}(M)=\text{null}(\text{rref}(M))$