Image - Maths

The image of a linear transformation is the subspace consisting of all the transformed vectors from the original domain. More plainly

Image ^formula

Let $T:U\to V$ be a linear transformation. The image(or range) of $T$ is given by:

$$\text{img}(T)=\{\vec{v}\in V|\vec{v}=T(\vec{u})\text{ for some }\vec{u}\in U\}$$

The image is a subspace of $V$. The rank of $T$ is the dimension of the image:

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

Obtaining the Image §

Let $T:U\to V$ and let $\mathcal{B},\mathcal{C}$ be the ordered standard bases of $U$ and $V$, respectively.

1. We can find the transformation matrix of a linear transformation, $T$, given by:

2. Find the column space of $[T]$. In other words, find

$$\text{span}(\text{column vectors of }[T])$$

3. Because the transformation matrix is written as a matrix of coordinate vectors, this span is that of coordinate vectors , not the ‘full’ vectors.
4. Simply multiply the coordinate vectors by the basis vectors:

$$\vec{v}=[\vec{v}{]}_{\mathcal{C}}\cdot \mathcal{C}$$

5. The image is the set of these vectors, i.e. a subspace of $V$

$$\text{img}(T)=\{{\vec{v}}_1,{\vec{v}}_2,\dots {\vec{v}}_n\}$$