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Problem of the Week #69 - September 23rd, 2013

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Chris L T521

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Jan 26, 2012
995
Here's this week's problem.

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Problem: Suppose we are given an exact sequence of finite dimensional $K$-vector spaces and $K$-linear maps:
\[0\rightarrow V_1\rightarrow V_2\rightarrow\cdots\rightarrow V_n\rightarrow 0.\]
Prove that
\[\sum\limits_{i=1}^n (-1)^i\dim(V_i) = 0.\]

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Hint:
Use induction on $n$. Note that if $V_1\xrightarrow{\phantom{xx}\phi_1\phantom{xx}}{}V_2\xrightarrow{\phantom{xx}\phi_2\phantom{xx}}{}V_3\xrightarrow{\phantom{xx}\phi_3\phantom{xx}}{}V_4$ is exact, then so is $V_1\rightarrow V_2\rightarrow \ker(\phi_3)\rightarrow 0$.


Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Chris L T521

Well-known member
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Jan 26, 2012
995
This week's problem was correctly answered by johng. You can find his solution below.

I see no need to induct. Here's my solution:

 
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