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two sets generate the same vector space

suvadip

Member
Feb 21, 2013
69
Show that the sets \(\displaystyle \{a,b\}\) and \(\displaystyle \{a, b, a-b\}\) of real vectors generate the same vector space.

How to proceed with it? I guess the following expression is helpful.

\(\displaystyle c1*a+c2*b+c3*(a-b)=(c1+c3)*a+(c2-c3)*b=k1*a+k2*b\)
 

Sudharaka

Well-known member
MHB Math Helper
Feb 5, 2012
1,621
Show that the sets \(\displaystyle \{a,b\}\) and \(\displaystyle \{a, b, a-b\}\) of real vectors generate the same vector space.

How to proceed with it? I guess the following expression is helpful.

\(\displaystyle c1*a+c2*b+c3*(a-b)=(c1+c3)*a+(c2-c3)*b=k1*a+k2*b\)
Hi suvadip, :)

You have to show any vector taken from $<a,\,b>$ lies in $<a,\,b,\,a-b>$ and vice versa. If you take any vector $v\in <a,\,b>$ it is clear that $x\in <a,\,b,\,a-b>$ since every vector generated by $a$ and $b$ is in $<a,\,b,\,a-b>$. Conversely if you take any vector $v\in<a,\,b,\,a-b>$ then $v=k_1 a+k_2 b+k_3 (a-b)=(k_1+k_3)a+b(k_2-k_3)\in <a,\,b>$. Therefore, $<a,\,b>=<a,\,b,\,a-b>$.