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 $V$?