- #1
losiu99
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Hello! I have just another problem I can't figure out how to solve:
Consider a homomorphism of short exact sequences (it's all vector spaces):
[PLAIN]http://img814.imageshack.us/img814/9568/seq.png
Prove that:
(1) [tex]\sigma[/tex] is surjective iff [tex]\rho[/tex] is injective.
(2) [tex]\sigma[/tex] is injective iff [tex]\rho[/tex] is surjective.
Earlier parts of the exercise:
(1) [tex]\psi_2 (\hbox{Im } \sigma)=\hbox{Im } \tau[/tex]
There was also another,
(2) [tex]\phi_1(\ker \rho)=\ker \sigma[/tex],
but this is wrong, I'm afraid.
I'm deeply sorry, but I have no idea where to start.
Thanks in advance for any hints!
Homework Statement
Consider a homomorphism of short exact sequences (it's all vector spaces):
[PLAIN]http://img814.imageshack.us/img814/9568/seq.png
Prove that:
(1) [tex]\sigma[/tex] is surjective iff [tex]\rho[/tex] is injective.
(2) [tex]\sigma[/tex] is injective iff [tex]\rho[/tex] is surjective.
Homework Equations
Earlier parts of the exercise:
(1) [tex]\psi_2 (\hbox{Im } \sigma)=\hbox{Im } \tau[/tex]
There was also another,
(2) [tex]\phi_1(\ker \rho)=\ker \sigma[/tex],
but this is wrong, I'm afraid.
The Attempt at a Solution
I'm deeply sorry, but I have no idea where to start.
Thanks in advance for any hints!
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