81–90 of 95 posts
I just skimmed the article quickly. Are there other ways to learn about matrices? If you don't treat them as linear applications, they are just boring grids of numbers and the matrices multiplication doesn't make any sense.
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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.
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It's a nice article - you focus on matrices as a kind of operator that takes a vector as input and produces another vector. This is one side of the coin. 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 o…
That didn’t make any sense to, and I work with matrices every day. Are you trying to describe a dot product?
First think about row and column vectors. A row vector and a column vector can be combined via standard matrix multiplication to produce a real number. From that perspective, a row vector is a function that takes a column vector and returns a real number. Similarly, column vectors take row vectors as arguments and produce real numbers.
It turns out that row (column) vectors are the only linear functions on column (row) vectors. This result is known as the Reisz representation theorem. If I give you a linear function on a row vector, you can find a column vector so that computing my function is equivalent to calculating a matrix multiply with your column vector.
Now on to matrices. Matrices take one row vector and one column vector and produce a real number. I can feed a matrix a single argument - the row vector, say - so that it becomes a function that takes one more argument (the column vector) before it returns a real number. Sort of like currying in functional programming. But as we said, the only linear functions that map column vectors into real numbers are row vectors. So by feeding our matrix one row vector, we've produced another row vector. This is the "matrices transform vectors" perspective in the OP's article. But I think the "Matrices are linear functions" perspective is more general and more powerful.
This perspective of vectors, matrices, etc... as functions might seem needlessly convoluted. But I think it's the right way to think about these objects. Tricky concepts like the tensor product and vector space duality become relatively trivial once you come to see all these objects as functions.
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Notably though, shearings are very 'rare'. Any pertubation will make a shearing no longer a shearing. At least, if I remember correctly.
Same for the the orthogonal matrices, or the diagonal matrices, or the symmetric matrices, or the unit determinant matrices, or the singular matrices ... They are all sets of Lebesgue measure zero.
Singular matrices are special in the sense that they keep the matrix monoid from being a group. My category theory isn't strong enough to characterize it, but this probably also has a name.
Edit: I think the singular matrices are the 'kernel' of the right adjoint of the forgetful functor from the category of groups to the category of monoids. Though I must admit a lot of that sentence is my stringing together words I only vaguely know.
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Crikey, I hope I never have to talk to that mathematician! That's a terse, unintuitive definition that isn't very helpful unless you're already familiar with the concepts. (Also maybe you meant linear maps into bilinear?) 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 sys…
Math is just a much deeper topic than most others. The things people do in research level math can take a really long time to explain to a lay person because of the many layers of abstraction involved.
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Crikey, I hope I never have to talk to that mathematician! That's a terse, unintuitive definition that isn't very helpful unless you're already familiar with the concepts. (Also maybe you meant linear maps into bilinear?) 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 sys…
No, it turns bilinear maps into linear one! If you have three R-modules (one can read K-vector spaces if unfamiliar with modules) N,M,P and a bilinear map N×M→P then there is a unique linear map N⊗M→P compatible with the map N×M→N⊗M which is part of the structure of a tensor product. (What's really going on here in fancy terms is the so called Hom-Tensor adjunction because the _⊗M functor is adjoint to the Hom(M,_) f…
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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.
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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A metric is a distance function. Defining a metric on a space is one of ways you create a topology. 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 t…
In differential geometry the metric [1] is a tensor that defines the relationship of vectors in the space to vectors in the tangent space. The identity function as a metric means that you are in a locally flat space where geodesics (the path taken by traveling in a given direction) are straight lines. A metric in a traditional metric space is a global distance function; you can use the metric tensor in a Riemannian m…