document.write( "Question 455730: Let v1 and v2 be non zero vectors in a vector space V. Show that the following statements are equivalent.\r
\n" ); document.write( "\n" ); document.write( "a) v1 is not in Rv2=span{v2}\r
\n" ); document.write( "\n" ); document.write( "b)Rv1 doesn't equals Rv2 \n" ); document.write( "

Algebra.Com's Answer #312878 by robertb(5830) $\"\"$ $\"About$

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(a)==>(b): $\"v%5B1%5D\"$ not in $\"span%28v%5B2%5D%29\"$ ==> $\"v%5B1%5D+%3C%3E+kv%5B2%5D\"$ for any non-zero real number k.
\n" ); document.write( "==> For any non-zero real number m, $\"mv%5B1%5D+%3C%3E+mkv%5B2%5D\"$ , and hence no element of span{v1} can be found in span{v2}.\r
\n" ); document.write( "\n" ); document.write( "(b) ==> (a): $\"span%28v%5B1%5D%29+%3C%3E+span%28v%5B2%5D%29\"$ ==> there is element $\"kv%5B1%5D\"$ in $\"span%28v%5B1%5D%29\"$ that is not in $\"span%28v%5B2%5D%29\"$ , or ,
\n" ); document.write( " $\"kv%5B1%5D+%3C%3E+r+v%5B2%5D\"$ for any non-zero real number r, or
\n" ); document.write( " $\"v%5B1%5D+%3C%3E+%28r%2Fk%29+v%5B2%5D+=+qv%5B2%5D\"$ . Hence v1 is not in span{v2}. \n" ); document.write( "

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