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 Decomposition gets a lot more interesting once you know some basic Algebra. I suppose Linear can't get too abstract because non math majors have to take it, but starting from generalized ideas of transformations is a far better way to teach it imo.
Relearning Matrices as Linear Functions
31–40 of 95 posts
Re: Relearning Matrices as Linear Functions
#32Hey 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…
As a layman, the word "tensor" always intimidated me. As a programmer, I was surprised then when I found out that a tensor is just a multi-dimensional array (where the number of dimensions can be as small as 0). That was a concept I was already quite comfortable with.
Re: Relearning Matrices as Linear Functions
#33Earlier quoted context omitted.
>geometrically, all linear maps can be thought of as rotations and scalings. and reflections.
Shearings cannot be represented in this way.
They can if you add a dimension to the space. That's one of the reasons 3d graphics use 4d vectors and matrices.
Re: Relearning Matrices as Linear Functions
#34In 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 numbers and composing affine transformations.
To this day I think that was one of the most beautiful and eye opening lessons I've had in mathematics.
In hindsight, I think I got lucky that the teachers who wrote the curriculum this way were math, physics, and comp sci masters/phd's who looked at their own educations and decided that geometry class was a great Trojan horse for linear algebra.
Re: Relearning Matrices as Linear Functions
#35Re: Relearning Matrices as Linear Functions
#36Earlier quoted context omitted.
Or pretty much any other Linear Algebra book.
Axler's book has the advantage of skipping determinants in order to provide a more intuitive approach to linear algebra.
I strongly disagree skipping determinants provides a more intuitive approach to linear algebra. I don't know your background, but I'd venture a guess you feel it does because the Laplace expansion formula for computing the determinant[1] feels uninspired and out of place.
The reason determinants are hard to teach (in my opinion) is because a rigorous derivation of their formula isn't possible without first teaching multilinear algebra and constructing the exterior algebra. Once you do those things, the natural geometric interpretation of the determinant basically falls onto your lap. But it's still very useful for e.g. computing eigenvalues and using the characteristic polynomial, so it's taught before that context can be formalized.
Professors shouldn't teach determinants in the context of matrices, at least not at first. That's heavily computation-focused, and the symbol pushing looks really unmotivated and strange to students. Instead they should teach the basis-free definition of determinants (i.e. focus on the linear map, not the matrix transformation representing the linear map for some basis). Then the determinant is "only" the volume of the image of the unit hypercube under the linear transformation, which is where the parallelepiped comes in. If the linear transformation is invertible, the unit hypercube is transformed from an n-dimensional cube into an n-dimensional parallelogram, from which you can geometrically see the way the linear map transforms the entire vector space it's defined over.
3Blue1Brown has a very good video on the geometry underlying the determinant[2]. For a more rigorous presentation which constructs the exterior algebra and derives the determinant formula using the wedge product, Noam Elkies has notes[3][4] for when he teaches Math 55A at Harvard. Incidentally Noam Elkies uses Axler's book, and while he obviously approves of it he's pretty upfront in asserting that the determinant should be taught anyway[5].
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2. https://www.youtube.com/watch?v=Ip3X9LOh2dk
3. http://www.math.harvard.edu/~elkies/M55a.10/p8.pdf
Re: Relearning Matrices as Linear Functions
#37Earlier quoted context omitted.
They can if you add a dimension to the space. That's one of the reasons 3d graphics use 4d vectors and matrices.
You're talking about translation.
No, I wasn't, but I did confuse the terms. Shear can be done without the extra dimension. Skew transforms require the extra dimension, as does translation.
Re: Relearning Matrices as Linear Functions
#38Hey 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…
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 and alright for modules after seeing enough algebra)
Re: Relearning Matrices as Linear Functions
#39Earlier quoted context omitted.
>geometrically, all linear maps can be thought of as rotations and scalings. and reflections.
Shearings cannot be represented in this way.
Notably though, shearings are very 'rare'. Any pertubation will make a shearing no longer a shearing. At least, if I remember correctly.
Re: Relearning Matrices as Linear Functions
#40Hey 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…
I always just thought of it as a thing that is indexable.