Determine the number of linear transformations from $V$ to $V$

Determine the number of linear transformations from $$V$$ to $$V$$ given the associated matrix of a linear transformation and the vectors that the linear transformation maps from $$V$$ to $$V$$.

Please, can somebody help me with this question? Is there any theory behind this on how to determine the number of linear transformations? I suppose, it must be connected with the rank of the matrix associated with the transformation and the dimension of the vector space $$V$$, but I am not sure.

Should I determine the dimension of all linear maps from $$V$$ to $$V$$?

• I don't understand the question. If you are given the matrix, that defines the transformation so there would only be one. Presumably the vectors it maps are all the vectors of $V$ Sep 22 '19 at 18:53
• @RossMillikan For example, I want to map from $R^4$ to $R^4$. I have $a,b,c,d$ vectors in $R^4$ on which I want to apply the linear map. As a result of which I get the associated matrix of linear transformation. Suppose, its rank is $3$. The question is: How can I determine the number of linear transformations from $R^4$ to $R^4$ given this information? Sep 22 '19 at 18:57

So if $$V$$ has $$b$$ number of different basis, we get $$b^2$$ as result.