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for lecture four uh but during the break of course um two things I to catch up on uh one the Overflow room reminded me that I should repeat anything you say because otherwise it doesn't get transmitted or recorded so all or the speaking now to the Overflow room you just have to ask the question louder uh from for Mei remember just remind me real loud that I should repeat everything and the other thing that came up was that I hadn't drawn should have drawn the basic uh picture of uh that shows that the battery B is in this is this is our edge here that the battery B is in series with the uh resistor and the current source is in parallel so it's the this is It's the battery so the three parts of our our equation e equal B minus Au was Au comes from the voltage differences all that way B gives us the voltage difference across the battery so e is the voltage drop across the resistor and that's where ohms law operates so that's a figure I'm glad to ...

Yes, OK, four, three, two, one, OK, I see you guys are in a happy mood. I don't know if that means 18.06 is ending, or, the quiz was good. Uh, my birthday conference was going on at the time of the quiz, and in the conference, of course, everybody had to say nice things, but I was wondering, what would my 18.06 class be saying, because it was at the exactly the same time. But, what I know from the grades so far, they're basically close to, and maybe slightly above the grades that you got on quiz two. So, very satisfactory. And, then we have a final exam coming up, and today's lecture, as I told you by email, will be a first step in the review, and then on Wednesday I'll do all I can in reviewing the whole course. So my topic today is -- actually, this is a lecture I have never given before in this way, and it will -- well, four subspaces, that's certainly fundamental, and you know that, so I want to speak about left-inverses and right-inverses and then...