Using the rank-nullity theorem in a proof

**kevinlightman** · November 29th 2009, 08:59 AM

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

**HallsofIvy** · November 29th 2009, 06:54 PM

Originally Posted by kevinlightman

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

The first is pretty straight forward. In general, to prove " $A\subset B$ " you show: "if y is a member of A then y is a member or B".

Here you want to prove: $Im(S+ T)\subset Im(S)+ Im(T)$
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 $y_1= Sx$ and $y_2= Tx$ . Then $y_1$ is in Im(S), $y_2$ is in Im(T), and $y= y_1+ y_2$ . 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.

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