# help needed with a proof

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• Nov 27th 2006, 05:27 AM
Shaun Gill
help needed with a proof
Suppose that

M={(a b ;a,b,c belong to R
-b c)

and N={(x 0 ;x,y belong to R
y 0)

are subspaces of M2R. Show that (M:R) + (N:R) = ((M'intersect'N):R) + ((M+N):R)

Cheers, Shaun
• Nov 27th 2006, 08:47 AM
ThePerfectHacker
Quote:

Originally Posted by Shaun Gill
Suppose that

M={(a b ;a,b,c belong to R
-b c)

and N={(x 0 ;x,y belong to R
y 0)

are subspaces of M2R. Show that (M:R) + (N:R) = ((M'intersect'N):R) + ((M+N):R)

Cheers, Shaun

I presume,
\$\displaystyle M+N=M\cup B\$
This looks like the principle of inclusion-exlusion.
Thus,
\$\displaystyle |M + N|=|M|+|N|-|M\cap N|\$
Thus,
\$\displaystyle |M + N|+|M\cap N|=|M|+|N|\$
So dimensions are,
\$\displaystyle (M+N:R)+(M\cap N:R)=(M:R)+(N:R)\$
I believe that is the idea.

But the sad thing is I do not know how to generalize this to infinite fields. So far this confirms what you said for finite fields.
• Nov 27th 2006, 09:00 AM
Plato
I think that these are the same problem.
http://www.mathhelpforum.com/math-he...ons-basis.html