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Let $E_1$ and $E_2$ be projections on $V$, a vector space over $F$. Why is if $\operatorname{char}F\neq2$ then $E_1+E_2$ is a projection iff $E_1E_2=E_2E_1=0$ ?

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You are quite right, I apologize. I have not encountered char before, nor have been taught about it, so the wording of the question confused me. –  Freeman Aug 31 '11 at 16:18
    
@Asaf: the LaTeX improvement may have increased readability; however, in view of mt_'s comment, LHS had turned the question from a completely wrong statement into a correct one while the LaTeX edit re-incorporated the first version. –  t.b. Aug 31 '11 at 16:28
    
I rolled back because I was mid way through correcting it when you edited it, i'm very grateful for the edit, but it was the original incorrect version. –  Freeman Aug 31 '11 at 16:29
    
LHS:Have you tried just doing a "binomial" with $E_1+E_2$, and then using the identity $E_1E_2=E_2E_1=0$? –  gary Aug 31 '11 at 16:30
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@Asaf: I'm sorry, but I have never used LaTeX before. –  Freeman Aug 31 '11 at 16:35
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1 Answer

up vote 2 down vote accepted

Think about it this way: $E_1+E_2$ is a projection if it satisfies: $(E_1+E_2)^2=(E_1+E_2)$

(Use $E_iE_j$ to mean the composition)

1)Assume $E_1E_2=0$

We want to show that $(E_1+E_2)(E_1+E_2)=(E_1+E_2)$ This means that $E_2E_1+E_2E_2+.....=(E_1+E_2)$ Can you see the next step?

For the converse, assume $(E_1+E_2)$ is a projection, then it must satisfy $(E_1+E_2)^2=....$

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Ok, so I see if E1E2=E2E1=0 as this implies E1E1+E1E2+E2E1+E2E2=E1E1+E2E2, implying (E1+E2)^2=E1+E2 But for the converse you have E1+E2 as a projection.. i'm unsure how you can show E1E2=E2E1=0 –  Freeman Aug 31 '11 at 16:46
    
Where do we use the fact 1+1 is not equal to 0? –  Freeman Aug 31 '11 at 16:47
    
Actually, you're right; it seems all we need is $E_1E_2+E_2E_1=0$. Let me see... –  gary Aug 31 '11 at 16:49
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No problem; I have been kind of slow myself; If we assume $E_iE_j=0$, and we have $1+1=0$, then $(E+E)^2$=$E^2+EE+EE+E^2$=$E^2+0+0+E^2=2E^2=0$ –  gary Aug 31 '11 at 17:13
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Yes, I think that works; let me double-check. –  gary Aug 31 '11 at 17:25
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