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I'm through the first 16 exercises of the 1.3 problems. Probably won't find time for the rest for a couple of days. We definitely want to make sure that everyo

I'm through the first 16 exercises of the 1.3 problems. Probably won't find time for the rest for a couple of days. We definitely want to make sure that everyone gets the "Recommended" problem from 1.3. It's a very important concept.

gruseom 15y ago
Do you mean 1.3.18, which asks why it is impossible for a system of linear equations to have exactly two solutions? This is one where people would probably come up with a variety of different explanations. Algebraically, if X and Y are solutions, then aX + bY is a solution if a + b = 1. Geometrically, if two planes intersect at a line, and you try to add a third plane that shares two points with them, you're forced to "capture" the entire line. Comments and/or additional ways of looking at this?
- mechanical_fish 15y ago
  Yes, that's the one I mean, and those explanations pretty much capture it. I might phrase the first explanation more simply as: If X and Y are solutions, then the point halfway between X and Y (i.e. X/2 +Y/2) is a solution. Then you can deploy Zeno's "paradox" to conclude that there is in fact an entire line of solutions. ;) But, of course, your statement is the more general one. This is important because it captures the vital quality of linear systems, the quality that makes linear algebra worth studying in the first place: Once you've got more than one solution to a linear system, any weighted sum of those solutions is also a solution. This leads to the exciting possibility that you don't need to carefully seek out every single solution (of which there is often an infinite number); you just need to find a handful of (hopefully simple) solutions that are sufficient to produce any of the other solutions by constructing weighted sums. This is pretty much the fundamental strategy of physics: Fourier analysis is the classic example of that strategy at work, but it's also the basis of quantum mechanics.
  - gruseom 15y ago
    Fabulous comment. Makes me think you're sandbagging us with this "I never learned linear algebra" schtick :) I was musing about this off and on yesterday and another way of looking at it struck me. Linear systems are flexible in some (well, few) ways and rigid in others. The reason you can't build a model that just captures two points and then absconds without getting stuck with infinitely more is that, being linear, this stuff can't "bend". Another way of putting it (the sort of thing someone who has spent too much time around computing and not enough around geometry would say) is that the language of linear systems is expressive enough to express certain things but not others. The possible interactions are regimented. You can add dimensions to gain degrees of freedom, or equations to collapse them, but both "freedom" and "restriction" here have a pretty narrow meaning. I'm looking forward to learning about some of the magic things this theory can be made to do. It makes sense that something with such a regular, yet complex structure (complex by our standards anyway - mathematicians would no doubt chuckle) could yield some beautiful results, but that it found such a variety of real applications is surprising to me. More surprising in this respect than its older cousin, calculus, which starts off much closer to our naive sense of how reality works. Now for those matrix exercises.
tptacek 15y ago
I was supposed to be in NYC this week but pushed it out, so I think if you're going to be done with 1.3 problems in a couple days, I stand a good chance at catching up to you.

gruseom 15y ago

Do you mean 1.3.18, which asks why it is impossible for a system of linear equations to have exactly two solutions? This is one where people would probably come up with a variety of different explanations. Algebraically, if X and Y are solutions, then aX + bY is a solution if a + b = 1. Geometrically, if two planes intersect at a line, and you try to add a third plane that shares two points with them, you're forced to "capture" the entire line. Comments and/or additional ways of looking at this?

mechanical_fish 15y ago
Yes, that's the one I mean, and those explanations pretty much capture it. I might phrase the first explanation more simply as: If X and Y are solutions, then the point halfway between X and Y (i.e. X/2 +Y/2) is a solution. Then you can deploy Zeno's "paradox" to conclude that there is in fact an entire line of solutions. ;) But, of course, your statement is the more general one. This is important because it captures the vital quality of linear systems, the quality that makes linear algebra worth studying in the first place: Once you've got more than one solution to a linear system, any weighted sum of those solutions is also a solution. This leads to the exciting possibility that you don't need to carefully seek out every single solution (of which there is often an infinite number); you just need to find a handful of (hopefully simple) solutions that are sufficient to produce any of the other solutions by constructing weighted sums. This is pretty much the fundamental strategy of physics: Fourier analysis is the classic example of that strategy at work, but it's also the basis of quantum mechanics.
- gruseom 15y ago
  Fabulous comment. Makes me think you're sandbagging us with this "I never learned linear algebra" schtick :) I was musing about this off and on yesterday and another way of looking at it struck me. Linear systems are flexible in some (well, few) ways and rigid in others. The reason you can't build a model that just captures two points and then absconds without getting stuck with infinitely more is that, being linear, this stuff can't "bend". Another way of putting it (the sort of thing someone who has spent too much time around computing and not enough around geometry would say) is that the language of linear systems is expressive enough to express certain things but not others. The possible interactions are regimented. You can add dimensions to gain degrees of freedom, or equations to collapse them, but both "freedom" and "restriction" here have a pretty narrow meaning. I'm looking forward to learning about some of the magic things this theory can be made to do. It makes sense that something with such a regular, yet complex structure (complex by our standards anyway - mathematicians would no doubt chuckle) could yield some beautiful results, but that it found such a variety of real applications is surprising to me. More surprising in this respect than its older cousin, calculus, which starts off much closer to our naive sense of how reality works. Now for those matrix exercises.

tptacek 15y ago

I was supposed to be in NYC this week but pushed it out, so I think if you're going to be done with 1.3 problems in a couple days, I stand a good chance at catching up to you.