# basis is minimal spanning set

I know minimal spanning set is independent . But how do i prove that a spanning set which is linearly independent i.e a basis is minimal spanning set ? I tried to prove by contradiction by assuming that a set which is smaller and different than the basis set is a minimal set . But this way prove seems to be very lengthy. is it possible that a smaller set containing different independent vectors (other than those is S) is minimal spanning vector ?

• i tried 5 hours to solve this problem . – Flintoff Feb 15 '18 at 17:25
• this is not a homework or assignment problem – Flintoff Feb 15 '18 at 17:25
• Just from curiosity of my mind – Flintoff Feb 15 '18 at 17:26

## 1 Answer

Let $S$ be your basis. If it was not a minimal spanning set, there would be a $s\in S$ such that $\bigl\langle S\setminus\{s\}\bigr\rangle$ is the whole space. Then, $s$ can be written as a linear combination $\alpha_1s_1+\cdots+\alpha_ns_n$ of elements of $S\setminus\{s\}$. That is, we have$$1.s-\alpha_1s_1-\cdots-\alpha_ns_n=0,$$ which is impossible, since $S$ is linearly independent.

• but is it possible that a smaller set containing different independent vectors (other than those is S) is minimal spanning vector ? – Flintoff Feb 15 '18 at 17:39
• The cardinality of all the bases of a vector space is the same, so it is not possible. – Laura Feb 15 '18 at 17:44
• @neraj Are you assuming that $S$ is finite? – José Carlos Santos Feb 15 '18 at 17:44
{"ft_lang_label":"__label__en","ft_lang_prob":0.8839907,"math_prob":0.9917249,"size":1584,"snap":"2019-35-2019-39","text_gpt3_token_len":446,"char_repetition_ratio":0.1278481,"word_repetition_ratio":0.21052632,"special_character_ratio":0.29608586,"punctuation_ratio":0.0882353,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99717337,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-20T16:34:04Z\",\"WARC-Record-ID\":\"<urn:uuid:ba98146d-9b0a-4de9-a0d5-61c49893fd21>\",\"Content-Length\":\"138396\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:38a68662-f199-466a-a412-48c103804313>\",\"WARC-Concurrent-To\":\"<urn:uuid:2ee4d467-fbad-4329-ada7-52efed4fc225>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/2651944/basis-is-minimal-spanning-set\",\"WARC-Payload-Digest\":\"sha1:LY6OIZRGONQEL36LUGATK2TIO3H2PM4A\",\"WARC-Block-Digest\":\"sha1:6SFSFKV7YPGMDTPWOCFWKQBKZFCWGTGV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027315551.61_warc_CC-MAIN-20190820154633-20190820180633-00036.warc.gz\"}"}