Image

62. Image

The image (or range) of a linear transformation $𝑇 : ℝ^{𝑛} \to ℝ^{𝑚}$ is the set of all possible outputs:

im (𝑇) = 𝑇 (ℝ^{𝑛}) = {𝑇 (\vec{𝑥}) | \vec{𝑥} \in ℝ^{𝑛}}

This is a generalization of the image of a subset applied to the entire domain.

62.1. Matrix representation

If $𝑇$ is represented by an $𝑚 \times 𝑛$ matrix $𝐴$ with columns ${\vec{𝑎}}_{1}, \dots, {\vec{𝑎}}_{𝑛}$ , then for any input $\vec{𝑥} = (𝑥_{1}, \dots, 𝑥_{𝑛})$ :

𝑇 (\vec{𝑥}) = 𝐴 \vec{𝑥} = 𝑥_{1} {\vec{𝑎}}_{1} + 𝑥_{2} {\vec{𝑎}}_{2} + \dots + 𝑥_{𝑛} {\vec{𝑎}}_{𝑛}

— a linear combination of $𝐴$ ’s columns.

62.2. Image = Column space

The image of $𝑇$ equals the column space of $𝐴$ :

im (𝑇) = Col (𝐴) = span ({\vec{𝑎}}_{1}, {\vec{𝑎}}_{2}, \dots, {\vec{𝑎}}_{𝑛})

Example

𝐴 = [\begin{matrix} 2 & 1 \\ 1 & 3 \end{matrix}]

$𝑇 ([\begin{matrix} 1 \\ 0 \end{matrix}]) = [\begin{matrix} 2 \\ 1 \end{matrix}]$ , $𝑇 ([\begin{matrix} 0 \\ 1 \end{matrix}]) = [\begin{matrix} 1 \\ 3 \end{matrix}]$ .

im (𝑇) = span ([\begin{matrix} 2 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 3 \end{matrix}]) = ℝ^{2}

(The two columns are linearly independent, so they span $ℝ^{2}$ .)

62.3. Properties

$im (𝑇)$ is a subspace of the codomain $ℝ^{𝑚}$
$dim (im (𝑇)) = rank (𝐴) =$ rank of the matrix
Rank–Nullity: $dim (im (𝑇)) + dim (ker (𝑇)) = 𝑛$
$𝑇$ is surjective iff $im (𝑇) = ℝ^{𝑚}$ iff $𝐴$ has full row rank

62.4. Three names for the same thing

Image of $𝑇$ — emphasis on the linear-map perspective
Range of $𝑇$ — same, common in analysis
Column space of $𝐴$ — same, matrix-algebra perspective