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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 $V$?

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  • $\begingroup$ 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 $V$ $\endgroup$ Sep 22 '19 at 18:53
  • $\begingroup$ @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? $\endgroup$
    – user13
    Sep 22 '19 at 18:57
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The only way to get different linear transformations is two change basis of domain and range.

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

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