# What is an “Image” - Linear Transformation

Pretty simple and straightforward. I just need to clarify what an image really means. Is it simply the result of applying a transformation on a vector x ? Moreover, what does it mean to find a standard matrix; I fail to understand the reasoning behind applying the transformation on the standard basis e -- is it the conventional way to do it, does it always work?

• Do you understand set-builder notation? If so then the image is defined like this: Let $T: V \to W$ be a linear transformation, then $\operatorname{image}(T) = \{w \in W \mid w= T(v)\text{ for } v\in V\}$. The reason to find the standard matrix representing a transformation is that the standard basis is super easy to work with. For instance if $A$ is any matrix and $\mathbf e_i$ is the $i$th standard basis vector of $\Bbb R^{n\times 1}$ then $A\mathbf e_i$ is the $i$th column of $A$ (prove this for yourself). – user137731 Oct 28 '15 at 2:00
• Or in words: the set of all possible values for $T$, the image of the definition set through $T$. – mvw Oct 28 '15 at 2:05