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Relearning Matrices as Linear Functions

dhruvonmath.com

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Re: Relearning Matrices as Linear Functions

#71

Post by swiley »

_hardwaregeek wrote: post #31

Linear Algebra, at least at my school, is taught pretty poorly. Instead of teaching the beauty of transformations, the course is boggled down in numerical nonsense and tedious calculations (who wants to find the inverse of a 3x3 matrix? Bueller? Bueller?). Only after learning Algebra and homomorphisms, isomorphisms and automorphisms did I appreciate the importance of linear transformations. Stuff like Singular Value…

At mine we have an "applied" linear algebra course for engineering students and the "normal" one that math people take.

I didn't pass the "normal" one but I think that's because I had another 300 level math course, capstone, and four other CS courses at the same time. I'm certain I wouldn't have passed the applied one, it looked very tedious and that's usually what gets me with homework.

Re: Relearning Matrices as Linear Functions

#72

Post by chongli »

soVeryTired wrote: post #53
ginnungagap wrote: post #38

Earlier quoted context omitted.

If you ask a mathematician a tensor is an element of a tensor product, just like a vector is an element of a vector space. This moves the question to "what is a tensor product", which you can think about as a way to turn bilinear maps into linear maps (this is an informal statement of the universal property of the tensor product, you also need a proof of existence of such an object, but it's easy for vector spaces an…

Crikey, I hope I never have to talk to that mathematician! That's a terse, unintuitive definition that isn't very helpful unless you're already familiar with the concepts. (Also maybe you meant linear maps into bilinear?) Reminds me of the time an algebraist mentioned to me that he was working on profinite group theory. I asked what a profinite group was, and he immediately replied 'an inverse limit of an inverse sys…

Math is just a much deeper topic than most others. The things people do in research level math can take a really long time to explain to a lay person because of the many layers of abstraction involved.

Re: Relearning Matrices as Linear Functions

#73

Post by gunnihinn »

avip wrote: post #20
dhruvp wrote: post #15

Hey OP here! When I first was introduced to matrices (high school) it was in the context of systems of equations. Matrices were a shorthand for writing out the equations and happened to have interesting rules for addition etc. It took me a while to think about them as functions on their own right and not just tables. This post is my attempt to relearn them as functions which has helped me develop a much stronger intu…

>geometrically, all linear maps can be thought of as rotations and scalings. and reflections.

And projections. Without them you only get linear maps with nonzero determinant.

Re: Relearning Matrices as Linear Functions

#74

Post by rramadass »

munchbunny wrote: post #34

In my high school matrices were first taught in geometry class, starting with using matrices as affine transformations in 2-d and then 3-d, and using that to teach concepts like what eigenvectors/values are, the equivalence of matrix and function composition, etc. That was taught right after a unit on complex numbers and trigonometry so that we could see the parallels between composing polynomial functions on complex…

You certainly were lucky to be taught Linear Algebra in such a manner! I came to understand the importance of such an approach only after a lot of head-scratching and self-study. IMO, a beautiful and important branch of "Practical" Maths has been needlessly obscured by the pedantic formalism espoused by the teaching community. Linear Algebra SHOULD always be taught alongside Coordinate/Analytic Geometry and Trigonometry for proper intuition.

I found the book "Practical Linear Algebra: A Geometry Toolbox" very helpful in my study.

Re: Relearning Matrices as Linear Functions

#75

Post by ginnungagap »

soVeryTired wrote: post #53
ginnungagap wrote: post #38

Earlier quoted context omitted.

If you ask a mathematician a tensor is an element of a tensor product, just like a vector is an element of a vector space. This moves the question to "what is a tensor product", which you can think about as a way to turn bilinear maps into linear maps (this is an informal statement of the universal property of the tensor product, you also need a proof of existence of such an object, but it's easy for vector spaces an…

Crikey, I hope I never have to talk to that mathematician! That's a terse, unintuitive definition that isn't very helpful unless you're already familiar with the concepts. (Also maybe you meant linear maps into bilinear?) Reminds me of the time an algebraist mentioned to me that he was working on profinite group theory. I asked what a profinite group was, and he immediately replied 'an inverse limit of an inverse sys…

No, it turns bilinear maps into linear one! If you have three R-modules (one can read K-vector spaces if unfamiliar with modules) N,M,P and a bilinear map N×M→P then there is a unique linear map N⊗M→P compatible with the map N×M→N⊗M which is part of the structure of a tensor product. (What's really going on here in fancy terms is the so called Hom-Tensor adjunction because the _⊗M functor is adjoint to the Hom(M,_) functor, but just thinking about bilinear and linear maps is much clearer)

Re: Relearning Matrices as Linear Functions

#77

Post by joppy »

throwawaymath wrote: post #60
ijidak wrote: post #58

Earlier quoted context omitted.

The metric?

A metric is a distance function. Defining a metric on a space is one of ways you create a topology. I'm not sure what the parent means by the metric being the identity function, however. The Euclidean metric is basically the hypotenuse of a triangle parameterized by two vectors. The adjacent and opposite sides of the triangle are measured to be the Euclidean norm of each vector (their length), and the hypotenuse is t…

The metric in a vector space is a dot product. If you just have one vector space, it’s not that interesting then, but if you have many vector spaces all glued together (like a tangent space to a curved surface), then looking at how the dot product varies between nearby tangent spaces tells you a lot about the surface (Gaussian curvature and so on).

Re: Relearning Matrices as Linear Functions

#78

Post by dagw »

whatshisface wrote: post #2

FWIW, I was told that matrices are linear maps pretty early on in my education. Are there any college level linear algebra / matrix calculations courses that don't tell students about that?

Sure they'll tell you that in passing, but won't really explain what that actually means. Certainly Linear Algebra 1 at college was a lot of apply this method to this object (that we'll call a matrix) to calculate this thing called a determinant. Don't worry about what it is or what it represents, just check if it's zero or not. If it's not zero apply this other method to calculate its inverse. Repeat.

Re: Relearning Matrices as Linear Functions

#79

Post by meuk »

It recently occurred to me that if you use that matrices represent linear functions, you don't have to do tedious math to prove that matrix multiplication is associative (that is, (A * B) * C = A * (B * C), which allows us to write A * B * C without brackets, since it doesn't matter how we place the brackets anyway).

For a matrix M, denote f_M(x) = M * x. Then f_{A * B} = f_A(f_B(x)) so that f_{(A * B) * C} = f_{A * B}(f_C(x)) = f_A(f_B(f_C(x))) and also f_{A * (B * C)} = f_A(f_{B * C}(x)) = f_A(f_B(f_C(x))).

So f_{(A * B) * C} = f_{A * (B * C)} = f_A(f_B(f_C(x)))

Re: Relearning Matrices as Linear Functions

#80

Post by S4M »

I just skimmed the article quickly. Are there other ways to learn about matrices? If you don't treat them as linear applications, they are just boring grids of numbers and the matrices multiplication doesn't make any sense.
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