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I'm writing about matrices & it strikes me that the literal definition of a matrix is very not representative of its utility in math. "A matrix is an array of numbers..." "Serena Williams is a member of the phylum chordata." Not incorrect but somehow spectacularly not the point.
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p.s. No need to explain better descriptions of matrices to me. I do know what I'm doing here.
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"Not incorrect but somehow spectacularly not the point."
Beautifully stated. :) May I borrow this?
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I wonder if it's because a matrix represents so many different things that it's hard to pick one and they revert to the lowest common denominator?
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Very cool thread, as always. I look forward to seeing your version of this. I've called it: 19th century intellectual technology to allow for the seamless, organized execution of countless applications of the distributive law.
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Well, it looks like I might end up cutting it from the thing I'm working on now, but I'm thinking about writing something else about it in the future. I like your description.
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Somewhere in the intersection of “not incorrect” and “spectacularly not the point” is the exact niche I fill with most of my writing.
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I appreciate this. Matrices were no part of any of my curricula. So I searched about them. Got the rules. Thought "well, that's nice, but WHY?"
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Yep, that's a terrible "definition". A matrix is a representation of a linear transformation with respect to a choice of bases. Unfortunately, this generally doesn't make sense until one's gone through the grief of the " numerical" version :-(
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I some senses, “an environment or material in which something develops” might be a metaphorical match
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Absolutely - an array of numbers is merely one way to represent a matrix, but has nothing to do with what a matrix actually is!
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An m-by-n matrix with entries in a ring R is a function from the Cartesian product {1,…,m}x{1,…,n} to R.
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this is why "why don't matrices multiply elementwise" is such a common question. "coefficients of a system of linear eqs" is much better
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I spend a big chunk of my (upcoming) book all about this :) I call it the "data definition" of a mathematical concept. More than just matrices, but matrices have got to be the most egregious example.
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Yeah - I really wish I understood this earlier. Linear Algebra seems rather arbitrary when I first encountered it
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Matrices also form vector spaces :)
I am a woman interested in linear algebra, my geet repo. is here, github.com/mitaxim/Linear
Do you have a webpage ?
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The one advantage may be that you can then go on to draw a clear distinction between linear transformations vs their representation(s) as a dead grid of numbers.
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...and then one gets to tensors and the generic description becomes combinatorially less relevant!
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Yeah but a single number is a pretty amazing thing by itself. Imagine... a whole ARRAY of them!
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It's great for coord.xform but awful for generalization to multilinears (e.g. tensor). If only there were a graphic/pictorial interpretive picture for the whole family as good as the "basis transform" for matrices is!
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This is why it is so frustrating to teach about them to high school students.
But how can we define mathematical concepts by reference to some measure of utility, when we don't know what utility that concept may have until later !!! Better rigorous pure mathematical definition should be independent from whatever application concept may have
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I find myself cast in the role of a "disagreer" here. A matrix is just a rectangular array of numbers. A linear transformation is a function T: V -> W which respects the linear structure. The two are *not* the same and the distinction is important.
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true for all definitions, they tell what an object is.What they are for is another story. Why I need em like this yet another.
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