Linear Algebra Done Wrong [pdf]

http://www.math.brown.edu/~treil/papers/LADW/LADW.pdf

by TriinT • 17 years ago

35 12 17 years ago

math.brown.edu

aconbere• 17 years ago

While not explicitly mentioned in the introduction I think it's safe to say that the books title is a play off of the popular title "Linear algebra done right"

http://www.amazon.com/Linear-Algebra-Right-Sheldon-Axler/dp/...

Which is a pretty amazing text if you're delving in to the algebra side of linear algebra. Though I suspect significantly less useful than "Linear algebra done wrong".

Dilpil• 17 years ago

I am always amazed at textbooks that introduce axioms and then later (or never!) show why they are interesting. Do any mathematicians actually think in this way?

wynand• 17 years ago

Not me. Well, I'm a post-grad and my Ph.D. is quite mathsy. That doesn't make me a mathematician (I still believe that the title should be earned and I definitely have not yet earned it), but at least I've done some mathematics.

Anyway, I always find it much easier to move from the concrete to the abstract. But once you've mastered an abstraction, it does seem to become "concrete" in your brain and then you can build on it. Thus, with a good background in basic linear algebra, you can move on to tensor analysis, but good luck if you want to jump into tensors straight away.

vitaminj• 17 years ago

To me, one of the interesting things about how mathematical results have been discovered historically is that they have often been accidental (or intended for something else).

eg. eigenvalues were originally looked at for use with quadratic forms, but people later found that its interesting properties (orthogonality, symmetry, etc) were useful for lots of other things.

There is often also a large gap in time between when a mathematician playing around with a problem discovers an interesting result and when it is actually applied to something useful.

eg. Euler's law regarding complex exponentials was developed around 1740, but it wasn't until around 1807 when Fourier used it in harmonic analysis and 1897 when Steinmetz started applying it to electrical engineering.

But you wouldn't know that from reading a textbook, which is (understandably) arranged in such a way that omits historical context and why people bothered to study it in the first place. Most linear algebra books are classic examples of how to introduce abstract topics without any context.

roundsquare• 17 years ago

My abstract algebra book from college introduced the axioms and used them right away. But new concepts were introduced independently and afterwards related to the axioms.

klodolph• 17 years ago

Looks suspiciously like the professor's notes from the linear algebra class I took (on the opposite side of the same country) for the first part. Then my professor started talking about homomorphisms, and things started getting GOOD.

A vector space is an abelian group, a scalar field, and a homomorphism from the field to automorphisms of the group. That's all you need to remember for definitions.

antimora• 17 years ago

Curious: has any of you used Linear Algebra in outside settings of academia? I know there are lots of uses in stats, numerical analysis, non-linear dynamics, and optimization but I've never used it practically.

corysama• 17 years ago

3D game development involves a lot of very basic linear algebra (small matrix operations) and the occasional application of advanced linear algebra (conjugate gradients and similar solvers).

hyperbovine• 17 years ago

Yes. It's huge in engineering, for example: http://commons.bcit.ca/math/examples/civil/linear_algebra/in...

According to one of my linear algebra professors from back when, a lot of computational linear algebra was developed by the Soviets in the mid-20th century as a way to carry out optimized central planning. So I guess another application would be, socialism.

No idea if this is true or not but it's a good story. Also goes a long way towards explaining why so many linear algebra textbooks are translated from Russian.

TriinT• 17 years ago

"Also goes a long way towards explaining why so many linear algebra textbooks are translated from Russian."

I think you're detecting a false pattern there. Russians generally do kick ass in Mathematics. In the West, Mathematics was held back by the Bourbaki fanatics, while in Russia they were never afraid of marrying the pure with the applied, the beautiful with the useful.

There may be a lot of Linear Algebra books translated from Russian, but there are also a lot of other books by Arnold, Kolmogorov, Fomin, Gelfand, etc that were also translated from Russian and that were not on Linear Algebra.

protez• 17 years ago

Linear algebra matters since we always prefer linearity to any other forms because of its simplicity. If something isn't linear, we should transform it into a linear form for further examination. That's what calculus does. That's what analysis does. That's the way how we think.

BrentRitterbeck• 17 years ago

Here's an example from the finance world. Let's say that you can't get a closed form solution for some option you are trying to price. One of the methods you may turn to is a finite difference method. You are going to have a series of equations to solve for the value at each grid point. A good understanding of linear algebra is essential if you want to code up your solution. You will be dealing with sparse matrices and tridiagonal matrices.

Linear Algebra Done Wrong [pdf]

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