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As with most things in mathematics, the varying perspectives on an object and the relationship between those perspectives are more important than what an object
by jfarmer 11y ago
As with most things in mathematics, the varying perspectives on an object and the relationship between those perspectives are more important than what an object is per se, to the point where the idea of what an object "is" per se is often meaningless.
Are the real numbers "actually" Dedekind cuts or equivalence classes of Cauchy sequences? If we prove that both constructions result in isomorphic objects, what difference does it make? Once equivalence has been established, we're free to adopt either perspective as the situation warrants.
Matrices represent linear transformations whether we want them to or not. As someone else pointed out, the operation you've defined is the Hadamard product[1], which is totally valid but doesn't correspond to the composition of linear transformations.
There are other "products", too, like Kronecker product[2] and the Frobenius product[3], each with the own properties, motivations, and relationships to other parts of mathematics. These are neither good nor bad nor anything else — they just are.
I think it was a misstep for the article to be titled What Matrices Are, because the real idea is that when we think of matrices as representing linear functions then the formula for "standard" matrix multiplication corresponds to the composition of linear functions. It's not just some crazy scheme we invented to torture Algebra II students in high school, but a different perspective on the composition operation that has its own advantages and disadvantages relative to other perspectives.
[1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices)
[2] https://en.wikipedia.org/wiki/Kronecker_product
[3] http://planetmath.org/frobeniusproduct
- jameshart 11y agoI think I'd be more comfortable with this article making the claim that linear transformation composition is "What matrix multiplication is". Because really, that's a more defensible position. If I weren't treating my matrices as representations of linear functions, I'd really have little reason to define the matrix multiplication operation that we all know and love - it's not a particularly useful operation on a rectangular array in general. So I guess, if you consider 'matrix multiplication' to be part and parcel of matrices, then sure - matrices are linear functions. And yes, understanding that is very important to motivate high school students. Affine transformations provide a good context for that motivation, as well as a good framework for intuiting noncommutativity of multiplication.
- jfarmer 11y agoTo be honest, I'm not sure what point you're trying to make. It feels like you're over-interpreting the title because the author uses much more precise language in the article. From the first paragraph, where he explains the purpose of the article: > The two fundamental facts about matrices is that every matrix represents some linear function, and every linear function is represented by a matrix. Therefore, there is in fact a one-to-one correspondence between matrices and linear functions. We’ll show that multiplying matrices corresponds to composing the functions that they represent. And later: > The connection is that matrices are representations of linear transformations, and you can figure out how to write the matrix down by seeing how it acts on a basis. If there's a meaningful difference between "a matrix is a representation of a linear transformation" and "a matrix can be viewed as a representation of a linear transformation" it seems largely philosophical and, in any case, tangential to the author's stated goal of explaining why matrix "multiplication" is defined the way it is. Whether or not matrix multiplication is a "particularly useful operation on a rectangular array in general" boils down to a debate about what is or isn't useful to do with a rectangular array. I'll leave that to other folks with stronger opinions on the matter.