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**0123** Umm..let's check if I have undestood the entire thing rightly:

the image is the space geneated by the columns of A. The rank is the max number of linearly indipendent vectors, so the rank is the dimension of the image. Then, since the image is generated by the columns of the matrix, if these are vectors linearly independent then they are a basis of the image. Otherwise let's eliminate the dependent ones and the lefts will be the basis.

Am I wrong or have I understood your explaination?