- #1
Chris L T521
Gold Member
MHB
- 915
- 0
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: [sp]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 $V_1\rightarrow V_2\rightarrow \ker(\phi_3)\rightarrow 0$.[/sp]
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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: [sp]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 $V_1\rightarrow V_2\rightarrow \ker(\phi_3)\rightarrow 0$.[/sp]
