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?
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Relearning Matrices as Linear Functions
21–30 of 95 posts
Re: Relearning Matrices as Linear Functions
#22Hey 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.
A reflection is just a scaling with a negative factor.
Re: Relearning Matrices as Linear Functions
#23Re: Relearning Matrices as Linear Functions
#24The next relearning step is to construct the category where arrows are matrices...
Re: Relearning Matrices as Linear Functions
#25Gilbert Strang uses similar approach on his Linear Algebra lectures. Much more intuitive
Re: Relearning Matrices as Linear Functions
#26Hey 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.
Shearings cannot be represented in this way.
Re: Relearning Matrices as Linear Functions
#27The next relearning step is to construct the category where arrows are matrices...
Is it Vect?
Re: Relearning Matrices as Linear Functions
#28The next relearning step is to construct the category where arrows are matrices...
> The next relearning step is to construct the category where arrows are matrices...
Why not the category of vector spaces (morphisms are linear maps)?
Re: Relearning Matrices as Linear Functions
#29FWIW, 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.
Re: Relearning Matrices as Linear Functions
#30This is great and a nice mathematical approach to the ideas of matrices. Another great resource is 3blue1brown's essence of linear algebra:
https://www.3blue1brown.com/essence-of-linear-algebra-page
Math is Fun also has a nice writeup that explain matrix multiplication from a real world example of a bakery making pies and tracking costs: