The Calculus Trap (2005)
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The Calculus Trap (2005)
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Re: The Calculus Trap (2005)
#2A couple of good past discussions:
Re: The Calculus Trap (2005)
#3> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too."
That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra?
Re: The Calculus Trap (2005)
#4> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too." That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from…
> That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school level. Maybe even Linear Algebra?
I am deeply confused by a curriculum which separates Matrix Theory from Linear Algebra. The description in the Wikipedia category just barely helps:
https://en.wikipedia.org/wiki/Category:Matrix_theory
> Matrix theory is a branch of mathematics which is focused on study of matrices. Initially, it was a sub-branch of linear algebra, but soon it grew to cover subjects related to graph theory, algebra, combinatorics and statistics as well.
The University of Missouri has a Matrix Theory course:
https://www.math.missouri.edu/class/matrix-theory
> Basic properties of matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and Jordan normal forms. Introduction to writing proofs.
... which specifies a textbook:
> Linear Algebra with Applications (7th edition) by Steven J. Leon
... which deepens my confusion. If you're taking that course, how is it not an introductory Linear Algebra course?
And this MathOverflow answer obfuscates again:
https://mathoverflow.net/questions/11669/what-is-the-differe...
> Let me elaborate a little on what Steve Huntsman is talking about. A matrix is just a list of numbers, and you're allowed to add and multiply matrices by combining those numbers in a certain way. When you talk about matrices, you're allowed to talk about things like the entry in the 3rd row and 4th column, and so forth. In this setting, matrices are useful for representing things like transition probabilities in a Markov chain, where each entry indicates the probability of transitioning from one state to another. You can do lots of interesting numerical things with matrices, and these interesting numerical things are very important because matrices show up a lot in engineering and the sciences.
> In linear algebra, however, you instead talk about linear transformations, which are not (I cannot emphasize this enough) a list of numbers, although sometimes it is convenient to use a particular matrix to write down a linear transformation. The difference between a linear transformation and a matrix is not easy to grasp the first time you see it, and most people would be fine with conflating the two points of view. However, when you're given a linear transformation, you're not allowed to ask for things like the entry in its 3rd row and 4th column because questions like these depend on a choice of basis. Instead, you're only allowed to ask for things that don't depend on the basis, such as the rank, the trace, the determinant, or the set of eigenvalues. This point of view may seem unnecessarily restrictive, but it is fundamental to a deeper understanding of pure mathematics.
If I try to parse charitably, I come away with the idea that Matrix Theory is about matrices as a data structure, usable for many things outside the scope of Linear Algebra, where they're all about using matrices to represent linear transformations. It's the difference between a column of numbers on a shopping bill and a column of numbers which represents a vector in a space with a specified basis. Gotcha.
However, this answer directly contradicts what the University of Missouri calls Matrix Theory, which is so Linear Algebra they even use a Linear Algebra textbook. It also... I don't know, trivializes the field of Matrix Theory. So you can manipulate matrices. So what? They show up a lot because they're used to represent specific things. Is the course going to barely introduce a lot of specific things and then focus on the matrix representation? What a waste!
Re: The Calculus Trap (2005)
#5> That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too.
I took calculus last year (AB Calc BC, 10th grade), and I can say that my experience was certainly a counterexample. I did OK, but it was definitely a marked difference from "breezing through" algebra.
Re: The Calculus Trap (2005)
#6> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too." That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from…
> That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school…
I didn't say that Linear Algebra and Matrix Theory were separated. I said that Matrix theory hit me like a ton of bricks. I took Linear Algebra in college prior to that, obviously.
I further stated that I think Linear algebra might benefit students if taught earlier, in high school.
Re: The Calculus Trap (2005)
#7> "For an avid student with great skill in mathematics, rushing through the standard curriculum is not the best answer. That student who breezed unchallenged through algebra, geometry, and trigonometry, will breeze through calculus, too." That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from…
> That was me. I was great at calculus type things, but Matrix Theory hit me like a ton of bricks. I still have that text book, sitting on my other desk, staring menacingly at me from across the room; Matrix Analysis, Horn and Johnson. Geometry in High School gave me a taste, but would have been nice had we had available another proof based class in the math curriculum; Formal Logic or Discrete Maths at a high school…
I think it will serve better to think about it as a difference of emphasis, rather than a difference of category. Maybe the term has been introduced to make the subject seem more tangible to newcomers? I notice that University of Missouri doesn't also have a "linear algebra" course.
As you say, "Matrix Theory is about matrices as a data structure" ends up being a pretty hollow concept anyway, because the important part is always related to matrix multiplication. The example of Markov chain models only underlines that point, since the main results of that theory depend entirely on the transition function being linear, and not at all on whether we represent it by a matrix. To put it another way: matrix multiplication is composition of linear operators. You can't extricate the matrix-as-data concept from the other.
Re: The Calculus Trap (2005)
#8The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations.
For young students, a great introductory textbook is Calculus Made Easy. It is around 100 years old, and develops all the material using infinitesimals. Which is essentially modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems.
For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.
Re: The Calculus Trap (2005)
#9The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations. For young students, a great introductory textbook is Calculus Made Easy . It is around 100 years old, and develops all the material using infinitesimals. Which is essentially modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thou…
I don’t know about the rest of the world but in Quebec calculus is not taught with any epsilon-delta formalism. You don’t encounter that until mathematical analysis, which is in the curriculum of a very few majors in uni.
Re: The Calculus Trap (2005)
#10The way calculus is usually taught is a mess. A mix of epsilon-delta formalism, without adequate motivation, differentials and excessive focus on computations. For young students, a great introductory textbook is Calculus Made Easy . It is around 100 years old, and develops all the material using infinitesimals. Which is essentially modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thou…
Yes, _Calculus Made Easy_ is a classic. Here's link to Project Gutenberg pdf: https://www.gutenberg.org/files/33283/33283-pdf.pdf