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.
{"ft_lang_label":"__label__en","ft_lang_prob":0.89591825,"math_prob":0.99971837,"size":551,"snap":"2022-05-2022-21","text_gpt3_token_len":117,"char_repetition_ratio":0.17184643,"word_repetition_ratio":0.0,"special_character_ratio":0.21052632,"punctuation_ratio":0.0776699,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99998796,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-22T07:18:11Z\",\"WARC-Record-ID\":\"<urn:uuid:5aa2b85b-779d-444d-860f-bdd5f84a6da1>\",\"Content-Length\":\"135195\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:147c086b-a8eb-4bba-94f6-708e4d8158aa>\",\"WARC-Concurrent-To\":\"<urn:uuid:14e46b44-ea43-481f-a9d6-fdc883b2473b>\",\"WARC-IP-Address\":\"151.101.129.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/3365917/determine-the-number-of-linear-transformations-from-v-to-v\",\"WARC-Payload-Digest\":\"sha1:OQN5PI5IWG76SNU7H7KIRHTZUQE6QJH6\",\"WARC-Block-Digest\":\"sha1:73YKBTRQ6U2LWSPWKY4OWD6S5HFUF3MK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320303747.41_warc_CC-MAIN-20220122043216-20220122073216-00067.warc.gz\"}"}