I'm stuck on this proof, can anyone help?

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- Nov 29th 2009, 08:59 AMkevinlightmanUsing the rank-nullity theorem in a proof
I'm stuck on this proof, can anyone help?

http://i929.photobucket.com/albums/a...earAlgebra.gif - Nov 29th 2009, 06:54 PMHallsofIvy
The first is pretty straight forward. In general, to prove "$\displaystyle A\subset B$" you show: "if y is a member of A then y is a member or B".

Here you want to prove: $\displaystyle Im(S+ T)\subset Im(S)+ Im(T)$

If y is in Im(S+ T) then y= (S+ T)x for some x in V. But "(S+ T)x" is**defined**as Sx+ Tx so we have y= Sx+ Tx. Let $\displaystyle y_1= Sx$ and $\displaystyle y_2= Tx$. Then $\displaystyle y_1$ is in Im(S), $\displaystyle y_2$ is in Im(T), and $\displaystyle y= y_1+ y_2$. Therefore, y is in Im(S)+ Im(T). The rest of that should be straight forward.

The second may be even simpler. Saying "ST= 0" mean (ST)v= 0 for all v. But that is the same as S(Tv)= 0 so that Tv is in the kernel of S for all v.