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 Posted: Sat Jun 20, 2009 9:40 pm
My pedagogical bent is kicking up again, so I'm going to attempt a few short tutorials on various things. Of course, being me, they're probably not going to be terribly informative, nor will they be terribly comprehensible. But I'm going to make an attempt. Basically I'm going to be flooding this forum with half-baked junk in the name of mathematical education. First I will attempt to encapsulate basic linear algebra, the notion of vector spaces, matrices, and the general notion of linear transformations. Topics include eigenstuff, function spaces, and if possible tensor products and wedge products. I'll try not to assume too much beside the notion of the Cartesian plane, and analogize from there. Then I will attempt (but not very seriously) to explain multivariable calculus. In a flat space, multivar is really quite boring, so I can't say that my heart will be in this. I'll do a brief thing on Fourier transforms here, because it's important to so much stuff and gives very cool results. Then we get to group theory; specifically I wish to discuss finite groups, subgroups (and normal subgroups), quotients, a few isomorphism things, and then on to the classical Lie groups and the linear groups. Then I can talk about group representations for finite and Lie groups. At this point I should probably discuss algebras and the relation between Lie algebras and Lie groups, as well as the relation between the representations of such. I'd also like to be able to talk about the Cayley-Klein geometries, as well as projective and affine geometries both finite and continuous. This, however, might require pictorial aids that I am not in possession of. Having done all of this, I'll make yet another attempt to explain geometric curvature because the rubber-sheet bowling ball analogy really has some terrible holes in it. All of this will be done without exercises, and I'm not great at examples either. There is little guarantee that this will ever finish (I need to get back to the QM tutorial at some point, which I will feel a lot better about once I've done all of this junk since I'll have a foundation rather than just vague "know some calculus"). But having had to tell many people in the past "I don't remember which book I used to learn multivar", it would be nice to just say "read this thread very slowly and make sure you understand it all."
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Posted: Sat Jun 20, 2009 10:04 pm
Linear algebra: topics, ideas, notes. To be updated repeatedly.
-Vectors as sets of coordinates: analogies to the Cartesian plane, discussion of R^n, C^n
-Vectors as algebraic structures, connection between this definition and the coordinate one
-Subspaces
-Linear independence, spanning, bases, dimensionality
-Vectors as direction and magnitude
-Dot product, cross product
-Briefly: function spaces
-Matrices: addition, multiplication, transpose
-Matrices as linear transformations
-Change of basis formulae
-Briefly: symmetric, skew-symmetric, Hermitian, skew-Hermitian
-Matrix polynomials
-Trace: invariance under change of basis
-Determinants: multiplicativity, invariance under change of basis, expansion of minors
-Matrix inverse, demonstrate on 2x2
-Eigenstuff
-Calculating eigenvalues: characteristic, minimal polynomials
-Inner product: vector norms other than Euclidean. Generalize from dot product
-Matrix format
-Positive-definiteness
-Bilinear versus sesquilinear
-Adjoint, self-adjointness: needs examples
-Direct Sum: simple enough
-Tensor Product: perhaps more difficult to explain without making them think it's just multiplication
-Wedge Product: need to analogize cross product without damaging too much
General stuff:
Unsurprisingly, I underestimated the length of the individual posts.
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Posted: Sat Jun 27, 2009 2:06 am
Notational notes:
New words should be italicized, section headings in bold, post titles in bold size large.
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Posted: Mon Jul 13, 2009 8:30 pm
Some things that deserve menitoon: -- Cauchy-Schwarz inequality. -- The particular case of function spaces: functions from a vector V to its own ground field, aka functionals. These form a vector space in its own right--the dual space V'. For the finite-dimensional case you have going, this has a direct connection to be both matrix transpose and dot product, i.e., the dual u is the function " u·". Layra-chan While the dot product can be defined in Rⁿ, the cross product only really makes sense in the cases of R² and R³. I'm going to disagree there. For R², you're just embedding the plane in R³ and treating them as vectors in the latter, and taking the new component. So what's a cross product? Layra-chan So in two dimensions, (the absolute value of) the cross product of two vectors gives the area A of the parallelogram spanned by the two vectors. ... if we denote the angle between u and v as θ, with counterclockwise being positive and clockwise being negative, we get that ux v = ||u||·||v||sin(θ) Missing absolute value. But more importantly: -- If we don't require the cross product to give a vector in the same space, then there's a natural generalization of the three-dimensional cross product as the Hodge dual of the wedge product. -- If we take the parallelogram relationship as the defining property, then then as you say, this works in R² and R³. However, it also works in R⁷. Specifically: (1) In R², take x and y to be complex numbers. Then: xy = x× y + i( x· y). (2) In R³, take x and y to be quaternions of zero scalar part. Then: xy = - x· y + x× y. (3) In R⁷, take x and y to be octonions of zero scalar part. Then define x× y to be the imaginary part of xy under octonion multiplication. (The scalar part is the negative of the regular dot product.) -- If both are taken as requirements, then only R³ and R⁷ are acceptable, but not R². -- If we also require that the cross product forms a Lie algebra, then the cross product becomes a unique feature of three dimensions.
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Posted: Tue Jul 14, 2009 10:47 pm
The R⁷ cross product point is a good one. The point about generalizing the cross product via the Hodge dual is not something I want to get into just yet because then I'd have to define the Hodge dual, and in any event that's something I should probably deal with when I get into the more general products. I was specifically concerned with outlining the cross-products that are defined in most school texts. The R² version I brought up is sadly canon, and I do in fact have all the absolute values correct, so I can't just discard it. I think I will rearrange that section, however, so that I can include the remark that the two-dimensional case is just a projection from the three-dimensional one.
I shall insert the R⁷ case.
Where do you recommend mentioning the Cauchy-Schwarz inequality and functionals?
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Posted: Wed Jul 15, 2009 12:31 am
Layra-chan The R⁷ cross product point is a good one. The point about generalizing the cross product via the Hodge dual is not something I want to get into just yet because then I'd have to define the Hodge dual, and in any event that's something I should probably deal with when I get into the more general products. I understand that; it's that for me, it's very strange to have a cross product that doesn't return a vector in the same space. Layra-chan I was specifically concerned with outlining the cross-products that are defined in most school texts. The R² version I brought up is sadly canon, ... Not in my experience, but either way, it's not a big deal. I can see that it would make sense from several perspectives, and the correspondence to the complex:quaternion surprised ctonion chain is pretty nice (although the complex case is slightly different because of the scalar [real] part). Layra-chan ... and I do in fact have all the absolute values correct, so I can't just discard it. I meant here: That it should have been | ux v| (since that statement explicitly referred to your 2-dimensional case that gives a scalar, the norm is just an absolute value). Note that if θ is defined in terms of the dot product, then since the principal branch of the inverse cosine is [0,pi], sine wouldn't need an absolute value. But it's probably best to also put one there just to remove the need for such qualifications. Layra-chan Where do you recommend mentioning the Cauchy-Schwarz inequality and functionals? Given your current layout, functionals would have a natural introduction in the function space section, being a particular case of such. Rather than keeping that topic contained, however, it's probably best to split it up, unless you feel like a whole lot of rewriting. So my recommendation would be something like this: -- Definition in the function space section. -- Particular identification in the dot product section, i.e., vector --> covector, being a linear function that performs the dot product with a given vector. -- Some further elaboration in the matrix section, or whichever place you address the transpose notation you have going, and the notion of dual space. For the finite-dimensional real case, the components the identification of vector to covector is bijective and given by the transpose operation. As for Cauchy-Schwarz, the real case definitely in the dot product section, since it's basically equivalent to the statement that |cos θ| ≤ 1 or that the angle is always well-defined. But if you feel like it, perhaps the "abstract" section can be expanded into defining an inner product space that would have to cover the inequality as well. It may be worthwhile to at least mention dot product over the complex field as subtly different, involving a conjugate.
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Posted: Sat Jul 18, 2009 9:55 pm
The 2-d cross product is signed (for reasons not entirely clear to me) so I don't actually need the absolute value signs on the left side; the sign of the left side matches the sign of θ. I got rid of that part anyway, as it feels like clutter and detracts from the fact that the 2-d cross product is just the 3-d one flattened.
Actually, the most accurate analogue for the complex case would be 1-dimensional vectors getting mapped to purely imaginary numbers; then the real part would be -1 times the dot product (i.e. the product of the vectors interpreted as scalars) and the imaginary part would be the analogue of the cross product (i.e. 0).
Anyway, thanks for the advice about the Cauchy-Schwarz inequality and covectors.
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Posted: Thu Jul 23, 2009 9:05 pm
Layra-chan The 2-d cross product is signed (for reasons not entirely clear to me) so I don't actually need the absolute value signs on the left side; the sign of the left side matches the sign of θ. Hmm... you're right, for the principal branch of arcsine. I was thinking of defining θ in terms of the dot product, as is usually done in the abstract case, where the principal branch of arccosine would mismatch the signs (sin θ is always nonnegative there). As for it being signed in the first place, that's just a consequence of being directed. But enough about such a minor issue that doesn't even exist anymore... Layra-chan Actually, the most accurate analogue for the complex case would be 1-dimensional vectors getting mapped to purely imaginary numbers; then the real part would be -1 times the dot product (i.e. the product of the vectors interpreted as scalars) and the imaginary part would be the analogue of the cross product (i.e. 0). Yeah; being forced to play with the same dimensionality screws it up.
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