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For me, I don't think we should do the calculations by hand just for the sake of doing it. It should be done only as a means for understanding the core concept

For me, I don't think we should do the calculations by hand just for the sake of doing it. It should be done only as a means for understanding the core concepts. That is I much prefer to do exercises where the understanding of core concepts is brought to the attention. So the question is "What are the core concepts up to 1.3"? I'm sure singular vs. non-singular is one. Are pivots a core concept? I think open-ended questions in the 2nd half of 1.3 questions are good candidates for deeper understanding.

gruseom 15y ago
Are pivots a core concept? Good question. They don't feel that way to me. They seem more like the implementation detail of an algorithm (Gaussian elimination). Is the algorithm itself a core concept? Hard to say no to that, I guess, but I wonder. What about what Strang calls the "row picture" vs. the "column picture"? There you have two fundamentally different geometric interpretations of linear equations. That was a revelation to me. When I studied linear algebra a long time ago, I don't remember encountering the column interpretation. Perhaps I did and didn't pay attention; I was more interested in symbolic manipulation at the time, whereas now I want a sense of what the symbols mean geometrically.
- tptacek 15y ago
  The intro lecture on row/column pictures made this click for me better than the book did: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... 22:00 in.
- mechanical_fish 15y ago
  Amusingly, I'm in the entirely opposite boat. I managed to get an entire Ph.D. in solid state physics without taking an actual course in linear algebra (a procedure that I do not recommend, but everyone is young and foolish once). So I've got some idea of what vector spaces are about geometrically -- otherwise, graduate quantum mechanics would have been impossible to navigate -- but I'm really bad at manipulating the actual symbols. And, incidentally, I'm planning to attempt most of the problems. I've cherry-picked my way through this subject before, and I was ultimately displeased with the results. They say that the problem with an autodidact is that he has a fool for a teacher, and with linear algebra I learned that the hard way. Fortunately, now they've invented Open Courseware, Amazon.com, and HN, so I'll give the fool another chance.
  - gruseom 15y ago
    I propose that we explicitly mention any exercises that we stumble over. Yeah it may make us look like idiots but we're destined to look like idiots anyway, and there's a major upside: we're going to stumble over different things, which means we can provide good explanations to one another, and learn better. I have a few examples of this sitting at home but am hacking at the library at the moment. It is heavenly quiet here. I will post them later.
    - akkartik 15y ago
      Kinda half-baked aside: Sometimes I think it would be incredibly valuable to share our intermediate steps when working out solutions to problems. Kinda like how it's fascinating to see PG write that essay.
    - mechanical_fish 15y ago
      So far I'm stumbling over the part where I actually sit down and read the book and do the problems. ;) But my book only arrived last week, so I'm not feeling too guilty yet.
      - gruseom 15y ago
        One function of this group is to turn guilt into shame. :)
tptacek 15y ago
We probably don't need to overthink this too much. We're not giving out grades, so do as many exercises as you think are necessary. What would be awesome would be, as you decide which exercises to do, if you could tell the rest of us why you chose the ones you did. My maths suck, so I'm going to try to get through all of them.
- gruseom 15y ago
  I'm going to try to get through all of them That's inspiring. I've been working through them all so far, albeit slowly. It's worth noting that later chapters have fewer exercises. The one I'm looking forward to most, Chapter 8, has less than a third as many as Chapter 1 does. Presumably they get less mechanical as well. I'd be content if I find enough time to "pay my dues" the slow way through the earlier chapters. When I was younger I used to scorn the mechanical stuff. Now I'm taking pleasure in it. We'll see if that lasts. For those who don't intend to do all of them, here are some exercises I found interesting: 1.2: 3, 10, 11, 14, 15, 23 1.3: 4, 14, 17, 18 (that's about as far as I've got so far) (Also, see my comments above if you don't have a way to look up those exercises by number - and especially if you have the Indian edition.)
  - lepton 15y ago
    OK, these were interesting. Are we ready to share thoughts on them? For 1.2.10, I got a single equation involving the three unknowns--does that sound right?
    - gruseom 15y ago
      Let's talk in detail about 1.2 and hold off on 1.3 until tptacek confirms. Re 1.2.10, I think that's right. Because the x-coords grow by the same amount as you go from one point to the next, the y-coords must as well, giving y3-y2 = y2-y1.
      - mechanical_fish 15y ago
        Yes, this is correct. (Though I haven't looked to see which problem this is, I can pretty much tell it's that one.) If we're not through 1.3 yet I'm not far behind! Will try to plug away on my 1.2 problems in between bugfixes.
patrickyeon 15y ago
> For me, I don't think we should do the calculations by hand just for the sake of doing it. It should be done only as a means for understanding the core concepts. That's always been my take on the reason for doing hand calculations, but sometimes it takes much longer than I expect to really grasp matters. I've seen in really well-written math books some questions bo a much better job of leading you to discover concepts "on your own" than the teaching material. One of my favourite methods to learn is to write code to embody the lessons learnt. There's no better way to learn a concept than to develop it through to the end, imo.
- gruseom 15y ago
  It would be so much fun to work through the book by writing code to do all the interesting stuff. I've been dreaming about doing just this in K; I've wanted to properly learn K for years and this is a dream domain for it. Silentbicycle, who knows a lot about K, is planning to join in this group at some point too. But alas, I doubt I'll be able to do it. My main project takes up pretty much all my hacking energy and I won't allow anything to detract from that. I've never done side projects well. :(

gruseom 15y ago

Are pivots a core concept? Good question. They don't feel that way to me. They seem more like the implementation detail of an algorithm (Gaussian elimination). Is the algorithm itself a core concept? Hard to say no to that, I guess, but I wonder. What about what Strang calls the "row picture" vs. the "column picture"? There you have two fundamentally different geometric interpretations of linear equations. That was a revelation to me. When I studied linear algebra a long time ago, I don't remember encountering the column interpretation. Perhaps I did and didn't pay attention; I was more interested in symbolic manipulation at the time, whereas now I want a sense of what the symbols mean geometrically.

tptacek 15y ago
The intro lecture on row/column pictures made this click for me better than the book did: http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-1-the-geometry-of-linear-equations http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-... 22:00 in.
mechanical_fish 15y ago
Amusingly, I'm in the entirely opposite boat. I managed to get an entire Ph.D. in solid state physics without taking an actual course in linear algebra (a procedure that I do not recommend, but everyone is young and foolish once). So I've got some idea of what vector spaces are about geometrically -- otherwise, graduate quantum mechanics would have been impossible to navigate -- but I'm really bad at manipulating the actual symbols. And, incidentally, I'm planning to attempt most of the problems. I've cherry-picked my way through this subject before, and I was ultimately displeased with the results. They say that the problem with an autodidact is that he has a fool for a teacher, and with linear algebra I learned that the hard way. Fortunately, now they've invented Open Courseware, Amazon.com, and HN, so I'll give the fool another chance.
- gruseom 15y ago
  I propose that we explicitly mention any exercises that we stumble over. Yeah it may make us look like idiots but we're destined to look like idiots anyway, and there's a major upside: we're going to stumble over different things, which means we can provide good explanations to one another, and learn better. I have a few examples of this sitting at home but am hacking at the library at the moment. It is heavenly quiet here. I will post them later.
  - akkartik 15y ago
    Kinda half-baked aside: Sometimes I think it would be incredibly valuable to share our intermediate steps when working out solutions to problems. Kinda like how it's fascinating to see PG write that essay.
  - mechanical_fish 15y ago
    So far I'm stumbling over the part where I actually sit down and read the book and do the problems. ;) But my book only arrived last week, so I'm not feeling too guilty yet.
    - gruseom 15y ago
      One function of this group is to turn guilt into shame. :)

tptacek 15y ago

We probably don't need to overthink this too much. We're not giving out grades, so do as many exercises as you think are necessary. What would be awesome would be, as you decide which exercises to do, if you could tell the rest of us why you chose the ones you did. My maths suck, so I'm going to try to get through all of them.

gruseom 15y ago
I'm going to try to get through all of them That's inspiring. I've been working through them all so far, albeit slowly. It's worth noting that later chapters have fewer exercises. The one I'm looking forward to most, Chapter 8, has less than a third as many as Chapter 1 does. Presumably they get less mechanical as well. I'd be content if I find enough time to "pay my dues" the slow way through the earlier chapters. When I was younger I used to scorn the mechanical stuff. Now I'm taking pleasure in it. We'll see if that lasts. For those who don't intend to do all of them, here are some exercises I found interesting: 1.2: 3, 10, 11, 14, 15, 23 1.3: 4, 14, 17, 18 (that's about as far as I've got so far) (Also, see my comments above if you don't have a way to look up those exercises by number - and especially if you have the Indian edition.)
- lepton 15y ago
  OK, these were interesting. Are we ready to share thoughts on them? For 1.2.10, I got a single equation involving the three unknowns--does that sound right?
  - gruseom 15y ago
    Let's talk in detail about 1.2 and hold off on 1.3 until tptacek confirms. Re 1.2.10, I think that's right. Because the x-coords grow by the same amount as you go from one point to the next, the y-coords must as well, giving y3-y2 = y2-y1.
    - mechanical_fish 15y ago
      Yes, this is correct. (Though I haven't looked to see which problem this is, I can pretty much tell it's that one.) If we're not through 1.3 yet I'm not far behind! Will try to plug away on my 1.2 problems in between bugfixes.

patrickyeon 15y ago

> For me, I don't think we should do the calculations by hand just for the sake of doing it. It should be done only as a means for understanding the core concepts. That's always been my take on the reason for doing hand calculations, but sometimes it takes much longer than I expect to really grasp matters. I've seen in really well-written math books some questions bo a much better job of leading you to discover concepts "on your own" than the teaching material. One of my favourite methods to learn is to write code to embody the lessons learnt. There's no better way to learn a concept than to develop it through to the end, imo.

gruseom 15y ago
It would be so much fun to work through the book by writing code to do all the interesting stuff. I've been dreaming about doing just this in K; I've wanted to properly learn K for years and this is a dream domain for it. Silentbicycle, who knows a lot about K, is planning to join in this group at some point too. But alas, I doubt I'll be able to do it. My main project takes up pretty much all my hacking energy and I won't allow anything to detract from that. I've never done side projects well. :(