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"Linear algebra is the math of vectors and matrices." "The matrix-vector product is used to define the notion of a linear transformation" This is BS. * First:
by Garlef 12y ago
"Linear algebra is the math of vectors and matrices."
"The matrix-vector product is used to define the notion of a linear transformation"
This is BS.
* First: Linear Algebra actually is the math of linear transformations.
* Second: No sane mathematician would define linear transformations in terms of matrices.
Vectors and matrices are only one way to model linear transformations; They help beginners to imagine linear algebra; But often, refraining from using coordinates delivers a much clearer picture.
Maybe this should be called the "No BS guide to passing your engineering math tests".
- ivan_ah 12y agoI'm guessing you're reading the short tutorial, which starts off with some computational aspects of LA. As soon as I introduce linear transformations (a few sections later) I make an update to this statement turning it into "linear algebra is all about vectors and linear transformations."
- Garlef 12y agoYeah. I was only reading through the four pager.
- ams6110 12y agoIf the language of "sane mathematicians" was comprehensible to most people, there would be no need for books like this.
- tjradcliffe 12y agoWhat "math" means to people outside of math--including physicists and engineers--is completely different from what "math" means to people inside math. We really ought to have a different word for the two fields. Even "applied math" doesn't cut it, because most applied mathematicians still think like mathematicians, and it ain't like the rest of us. What mathematicians don't see is that when they rail against the kind of concrete, heuristic definitions that non-mathematicians use for parts of mathematics, is that it isn't just that we don't care that there is a better, more general, more abstract, more justifiable definition out there, it's that we can't understand them so they aren't useful to us, even a little bit. You may as well try explaining Shakespeare to your dog: it'll frustrate you and annoy the dog, who is perfectly happy to use Shakespeare for his intended purpose, which is to say, chewing on the edges of the "Complete Shakespeare" for entertainment. That's an overstatement, but the gulf between the mathematician's mind-set and the mind-set of physicists and engineers is still far wider than most mathematicians appreciate, and it really would be valuable to have a different word for what physicists and engineers do when we "do math", because it is not "math" the way you mean it.
- bkcooper 12y agoThat's an overstatement, but the gulf between the mathematician's mind-set and the mind-set of physicists and engineers is still far wider than most mathematicians appreciate, and it really would be valuable to have a different word for what physicists and engineers do when we "do math", because it is not "math" the way you mean it. I don't think that's true, at least in areas where physicists or engineers are actually going to be trying to use the relevant math. I think the difference is mostly in norms of written presentation. In physics, the concrete motivating example is often seen as an important thing to present; in math, it seems more typical to polish your definitions until they're as minimal and general as can be and start the presentation there. This often obscures the more concrete examples, but I've found plenty of times that if you read or talk to mathematicians in informal settings they can be pretty good about the examples that motivate the ideas.
- interdrift 12y agoCould you recommend a book for me so I can get deeper understanding?I'm a college student(software engineering). I love linear algebra,I passed it easly in my first semester, and I've done my studying and problems on Khan Academy.org .
- thearn4 12y agoIf you're interested in the numerical side, I'd recommend "Numerical Linear Algebra", by Trefethen and Bau.
- Garlef 12y ago"Linear Algebra Done Right": http://linear.axler.net/ http://linear.axler.net/ From a math perspective some definitions could be "cleaner" but as a first/second round it gives a very good perspective on linear algebra. Some of the good stuff: * The general theme is "direct sum decomposition". * Determinants are delayed as much as possible. (There are many good reasons to do this.) * Eigenvalues are not computed using the characteristic polynomial but the other way around; This reflects the algorithmic side: For performance reasons a computer usually calculates the char.polynomial using the eigenvalues and not by using determinants.
- wetmore 12y agoPick up a copy of Linear Algebra Done Right and start working through it. Edit: I see someone else linked to it in this comment thread.
- cbsmith 12y ago> Vectors and matrices are only one way to model linear transformations; They help beginners to imagine linear algebra; But often, refraining from using coordinates delivers a much clearer picture. They also make for a handy model when mapping them in to computer systems. It's hard to know from those quotes how things are actually presented in the material, but an emphasis on matrix & vector view of the world is really not a bad idea that I wish was more prevalent in math education.