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?
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!