Can someone help me understand @22:55~@23:59 (when the Professor starts the surjective proof of phi using x+y bar)? I got lost there, especially with the usage of bar notation for x+y. Thank you in advance!
He wants to prove that phi is surjective. A general element of the codomain looks like x+y+M_{1}, with x in M_{2} and y in M_{1} (it's a bit strange that he labels the variables this way, but that doesn't really matter). So he needs to find an element z in M_{2} with phi(z)=x+y+M_{1}. His working out at that point in the video is just to show that x+y+M_{1}=x+M_{1} (this is true because y is in M_{1}, so y+M_{1}=0). It follows that he can take z=x, i.e., phi(x)=x+y+M_{2}. This proves that phi is surjective.
Great lectures but the person taking the video is bad...you do not need to show the professors face all the time, sometimes focus on the board and not his face...we can listen to his voice without the face. Moving the camera all around makes me dizzy.
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Nice lecture! I personally love all pure math lecture series give at IIT on youtube!
given*
Excellent lectures thank you so much
Can someone help me understand @22:55~@23:59 (when the Professor starts the surjective proof of phi using x+y bar)? I got lost there, especially with the usage of bar notation for x+y. Thank you in advance!
He wants to prove that phi is surjective. A general element of the codomain looks like x+y+M_{1}, with x in M_{2} and y in M_{1} (it's a bit strange that he labels the variables this way, but that doesn't really matter). So he needs to find an element z in M_{2} with phi(z)=x+y+M_{1}. His working out at that point in the video is just to show that x+y+M_{1}=x+M_{1} (this is true because y is in M_{1}, so y+M_{1}=0). It follows that he can take z=x, i.e., phi(x)=x+y+M_{2}. This proves that phi is surjective.
@@wreynolds1995 Makes sense. Thank you for taking your time to answer my question!
thanks
Great lectures but the person taking the video is bad...you do not need to show the professors face all the time, sometimes focus on the board and not his face...we can listen to his voice without the face. Moving the camera all around makes me dizzy.
thanks