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I finished Ph.D. courses without ever having to pay much attention to the geometric interpretation of matrices. The path I took was solely computational and alg

by wenc 9y ago

I finished Ph.D. courses without ever having to pay much attention to the geometric interpretation of matrices. The path I took was solely computational and algebraic. I often wondered if I'd missed anything. The 3Blue1Brown videos say I did. I will admit I don't have a solid intuition for matrices used for rotations, flips, etc. (outside of basic 2x2 examples). This can be a disadvantage, even if one pursues a purely computational approach to linear algebra. For instance, it took me a little longer to understand numerical techniques like Householder transformations. https://en.wikipedia.org/wiki/Householder_transformation https://en.wikipedia.org/wiki/Householder_transformation That said, I want to also say something in defense of the purely algebraic approach. In higher dimensions, there is no geometric analogy and people who rely on geometry alone find themselves unable to think abstractly in higher dimensions. To me, matrices and vectors are the building blocks for multivariate generalizations of familiar relationships. The algebraic approach meant I was comfortable dealing with complicated high dimensional objects that weren't amenable to visualization. I believe intuitions are good for laying a foundation, but abstractions are necessary for generalization and for building larger theoretical systems. Intuitive constructs can only get you so far. If I'm not mistaken, this was what Bourbaki group strove for; that is, to move mathematics past the intuitionist approaches of the day to a solid theoretical foundation built upon abstractions. There are pros and cons, but arguably without this foundation, mathematics might not have grown in the way it did.

chii 9y ago
> Intuitive constructs can only get you so far. yes, but intuitive constructs helps the beginner. It makes the material easier to obsorb, and hopefully, by building this intuition up, it can lead to a deeper learning later on (what you call abstraction).
thethirdone 9y ago
> In higher dimensions, there is no geometric analogy and people who rely on geometry alone find themselves unable to think abstractly in higher dimensions. Often features in higher dimensions can start to be seen in lower dimensions. For example the common example of the strangeness of higher dimensional spaces is the case that a hypercube with side length 2 can have a smaller volume that a hypersphere touched by hyperspheres of radius 1 at each corner of the hypercube. This is not true in 3 or even 4 dimensional space. But if you use geometric intuition to see the trend from 1d to 2d to 3d, it seems entirely reasonable. (Apologies if I didn't explain the problem well. I wanted to find a video on it, but couldn't find one) > I believe intuitions are good for laying a foundation, but abstractions are necessary for generalization and for building larger theoretical systems. Intuitive constructs can only get you so far. I have a slightly different view on this. I feel that abstractions are fundamentally necessary for communicating rather than conceptualization. For example, you may realize that the type of theorems you have been making about vectors equally applies to polynomials, but in order to convince others you would need to write done the properties they both have (resulting in an abstraction). Its kind of a subtle point because communication is key to mathematics. But normally, before I try to write a proof, I figure out the intuition for why it must be true in my head. Because this is a powerful tool, I try to learn a solid intuition for any abstraction I am introduced to.
deleted 9y ago
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by wenc 9y ago

chii 9y ago

> Intuitive constructs can only get you so far. yes, but intuitive constructs helps the beginner. It makes the material easier to obsorb, and hopefully, by building this intuition up, it can lead to a deeper learning later on (what you call abstraction).

thethirdone 9y ago

> In higher dimensions, there is no geometric analogy and people who rely on geometry alone find themselves unable to think abstractly in higher dimensions. Often features in higher dimensions can start to be seen in lower dimensions. For example the common example of the strangeness of higher dimensional spaces is the case that a hypercube with side length 2 can have a smaller volume that a hypersphere touched by hyperspheres of radius 1 at each corner of the hypercube. This is not true in 3 or even 4 dimensional space. But if you use geometric intuition to see the trend from 1d to 2d to 3d, it seems entirely reasonable. (Apologies if I didn't explain the problem well. I wanted to find a video on it, but couldn't find one) > I believe intuitions are good for laying a foundation, but abstractions are necessary for generalization and for building larger theoretical systems. Intuitive constructs can only get you so far. I have a slightly different view on this. I feel that abstractions are fundamentally necessary for communicating rather than conceptualization. For example, you may realize that the type of theorems you have been making about vectors equally applies to polynomials, but in order to convince others you would need to write done the properties they both have (resulting in an abstraction). Its kind of a subtle point because communication is key to mathematics. But normally, before I try to write a proof, I figure out the intuition for why it must be true in my head. Because this is a powerful tool, I try to learn a solid intuition for any abstraction I am introduced to.

deleted 9y ago

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