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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?

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  • $\begingroup$ 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). $\endgroup$ – user137731 Oct 28 '15 at 2:00
  • $\begingroup$ Or in words: the set of all possible values for $T$, the image of the definition set through $T$. $\endgroup$ – mvw Oct 28 '15 at 2:05
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It's usually conventional to say that the "image" of x under a linear transformation is the effect of the transformation on vector x.

To find a standard matrix means to find a matrix that implements the map. It's reasonable to apply the transformation to the standard basis e because it's an orthonormal basis, something we like. So all you have to do is map the basis vectors, by applying the transformation to them. This would be your standard matrix. (eg. T( [1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1] ), for R^4). Hope this answers your question!

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