51–60 of 95 posts
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?
I'm sure I was told, but I don't think it was strongly emphasized by my instructors. It comes pretty late in Strang's text, for instance.
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…
What got me for a while was the concept of a tensor: For example: What is a tensor? Wrong way to answer it: Well, the number 5 is a tensor. So's a row vector. So's a column vector. So's the dot product and the cross product. So's a two-dimensional matrix. So's a four-dimensional matrix, just... don't ask me to write one on the board, eh? So's this Greek letter with smaller Greek letters arranged on its top right and…
A tensor is a function that takes an ordered set of N covariant vectors (i.e. row vectors) and M contravariant vectors (i.e. column vectors) and spits out a real number. It has to be linear in each of its arguments.
I'm pretty sure all the complicated transforms follow from that definition (though you may have to assume the Leibniz rule - I can't remember), and from ordinary calculus.
Earlier quoted context omitted.
What got me for a while was the concept of a tensor: For example: What is a tensor? Wrong way to answer it: Well, the number 5 is a tensor. So's a row vector. So's a column vector. So's the dot product and the cross product. So's a two-dimensional matrix. So's a four-dimensional matrix, just... don't ask me to write one on the board, eh? So's this Greek letter with smaller Greek letters arranged on its top right and…
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…
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 system', with no follow up. Well thanks buddy, that really opened my eyes.
If you're interested in this approach to linear algebra you should read Linear Algebra Done Right by Sheldon Axler.
Or pretty much any other Linear Algebra book.
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…
That was exactly my experience. Struggled with matrices theory at uni doing some bullshit exercises but started to grasp the topic only when I needed to apply some linear transformation in a game
[1]https://datasciencetexts.com/subjects/linear_algebra.html
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…
The other interpretation is that matrices are functions that take two arguments (a row vector and a column vector) and produce a real number. IMO this interpretation opens the door to deeper mathematics. It links in to the idea that a column vector is a functional on a row vector (and vice versa), giving you the notion of dual space, and ultimately leading on to differential forms. It also makes tensor analysis much more natural in general.
https://www.dhruvonmath.com/2019/04/04/kernels/
The matrix/function stuff is elementary enough that I understand it intuitively (I suck at math), although it's neat to be reminded that given a enough independent points you can reconstruct the function (this breaks a variety of bad ciphers, sometimes including ciphers that otherwise look strong).
The kernel post actually does some neat stuff with the kernel, which I found more intuitively accessible than (say) what Strang does with nullspaces.
It took until I started learning differential geometry in the form of General Relativity to arrive at this insight, even though I feel like the notion of a matrix as a linear map was drilled in pretty thoroughly. The notion of matrix multiplication as function composition was presented almost as an interesting side effect of matrix multiplication -- that is, multiplication by these rules came first, and, hey, look, t…
Having not taken a linear algebra course in college, does anyone have a recommendation for a book/course to follow?
Essence of Linear Algebra https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
It took until I started learning differential geometry in the form of General Relativity to arrive at this insight, even though I feel like the notion of a matrix as a linear map was drilled in pretty thoroughly. The notion of matrix multiplication as function composition was presented almost as an interesting side effect of matrix multiplication -- that is, multiplication by these rules came first, and, hey, look, t…
The metric?
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 the shortest distance between them.
The Euclidean metric is not the only metric - you can define distance however you'd like as long as it's consistent. But I'm not sure how the identity function works as a metric, because that would map a vector to another vector, not a scalar.