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The Little Book of Linear Algebra
121–130 of 134 posts
Re: The Little Book of Linear Algebra
#122Earlier quoted context omitted.
> Perhaps from graphics -- translation, rotation, reflection Maybe ... but the fact that you included translation in the list of linear operations seems like a big red flag. Translation feels very linear but it is emphatically not [1]. This is not intended to be a personal jab; just that the intuitions of linear algebra are not easy to internalize. Adding linear transformations is similarly scary territory. You can m…
Can you elaborate on your point that translation is not linear? The OP agrees with you, so clearly your point is correct, but I personally just don't understand it. Isn't it true that translation is linear within the coordinate space of your model, even if the final distance traveled within a projected camera view is not? edit to add: (I think your point relates only to the projection system, and not a pure, unprojec…
No, with projective geometry or affine geometry you can make translation into a linear operation. But in ordinary Euclidean space translation is not a linear operation.
Most obvious case that it fails is that it doesn't map zero to itself, and you can see the contradiction there:
T(0 + 0) = T(0) = t
T(0) + T(0) = t + t = 2 * tRe: The Little Book of Linear Algebra
#123It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…
The older I get the more convinced I am that "math is not hard; teaching math is hard".
As a maths teacher who is interested in (and sufficiently skilled at) programming, I find teaching programming to be very hard, even to interested students.
Teaching maths to interested students is not hard (for me).
Re: The Little Book of Linear Algebra
#124Earlier quoted context omitted.
> Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away In my experience it need not be like that at all. One can start by defining and demonstrating linear transformations. Perhaps from graphics -- translation, rotation, reflection etc. Show the students that these fol…
If anybody is aware of materials that teach linear algebra via graphics as suggested here, I would be interested to hear about them. As someone who learns best through practical application, maths have been by far among my greatest weak points, despite having written software for upwards of a decade. It’s limiting in some scenarios and pure imposter syndrome fuel.
Re: The Little Book of Linear Algebra
#125In terms of understanding why something is like that, Linear Algebra belongs to Geometry more than Algebra. Every formula in Linear Algebra which ultimately is justified by "it's just that way" can be better justified by geometry. The algebraic formulas are like animals who lost their natural habitat and were put in a zoo, making people think that these animals evolved in the zoo itself. To give an example: A simple…
Not just Linear Algebra but every branch of Science/Mathematics should be taught as much as possible using Geometry (and other visualizations) before being mapped to abstract algebraic symbols. Basic Geometry (along with simple Arithmetic) is the oldest subfield of Mathematics for the reason that most of its concepts are intuitive and feels "natural" and mappable to the "Real World" by us Humans. While abstraction via symbol manipulation is necessary to generalize and extend mathematics it should come at a later stage after we have developed some intuition of the concepts being represented by the symbols, however limited/restricted they might be. All abstraction requires some mathematical maturity which can only happen over time.
Re: The Little Book of Linear Algebra
#126Earlier quoted context omitted.
What I find amazing is, given how important linear algebra is to actual practical applications, high school math still goes so deep on calculus at the expense of really covering even basic vectors and matrices. Where vectors do come up it’s usually only Cartesian vectors for mechanics, and only basic addition, scalar multiplication and component decomposition are talked about - even dot products are likely ignored.
I think it was a brilliant and evil trick by the linear algebra folks. Start the path at calculus. Naturally, this will lead to differential equations. Trick the engineers into defining everything in terms of differential equations. The engineers will get really annoyed, because solving differential equations is impossible. Then, the mathematicians swoop in with the idea of discretizing everything and using linear al…
Ha, Ha, True dat :-)
Re: The Little Book of Linear Algebra
#127It's crazy that Linear Algebra is one of the deepest and most interesting areas of mathematics, with applications in almost every field of mathematics itself plus having practical applications in almost every quantitative field that uses math. But it is SOOO boring to learn the basic mechanics. There's almost no way to sugar coat it either; you have to learn the basics of vectors and scalars and dot products and matr…
>Even the "why does matrix multiplication look that way" is incredibly deep but practically impossible to motivate from other considerations. You just start with "well that's the way it is" and grind away until one day when you're looking at a chain of linear transformations you realize that everything clicks.
Can you expand on your experience with this? I do some graphics programming so I understand that applying matrix transformations works, and I've seen the 3blue1brown 'matrices are spreadsheets' explanation (luv me sum spreadsheets), but the intuition still isn't really there. The 'incredibly deep "why matrix multiplication looks that way"' is totally lost on me.
Re: The Little Book of Linear Algebra
#128Earlier quoted context omitted.
I dont agree with this. Matrices don't convert sets of basis vectors to sets of basis vectors. What would you say about singular matrices for example? The natural motivation of matrices is as representing systems of equations.
There's no single best way to understand any of this, but the action of a matrix on the standard basis vectors is a totally reasonable place to start because of its simplicity, and then the action on any vector can be built out of that because they're linear combinations of basis vectors.
i don't agree because this seems circular. You cant even define a matrix as something that acts on vectors meaningfully until you have some machinery.
if you start with a set S and then make it vector space V over field K. Then by definition, linear combinations (and its not an algebra so nonlinear isn't even defined) are closed in V.
You can then define spanning sets and linear independence to get bases. From bases you can define coordinate vectors over K^n as isomorphic to V. Then given some linear function f : V->W by definition f(v) = f(v^i * b_i) = v^i * f(b_i)
Only here is when you can even define a matrix meaningfully as a tuple of coordinate vectors which are the image of some basis vectors.
Then you need to prove that what was function application of linear functions on vectors is the same as a new operation of multiplication of matrices with coordinate vectors.
And then to prove the multiplication rule (which is inherently coordinate based) you are going make the same argument I made in sibling comment. But I could prove the rule directly by substitution using only systems of linear equations as the starting point.
Re: The Little Book of Linear Algebra
#129Earlier quoted context omitted.
There's no single best way to understand any of this, but the action of a matrix on the standard basis vectors is a totally reasonable place to start because of its simplicity, and then the action on any vector can be built out of that because they're linear combinations of basis vectors.
i don't agree because this seems circular. You cant even define a matrix as something that acts on vectors meaningfully until you have some machinery. if you start with a set S and then make it vector space V over field K. Then by definition, linear combinations (and its not an algebra so nonlinear isn't even defined) are closed in V. You can then define spanning sets and linear independence to get bases. From bases…
Where's the circularity?
What you're saying is fine as an abstract presentation, but I was talking about how students might initially come to learn about matrices, so just introducing column vectors as representing points in 2 and 3 dimensional space and how matrices transform them is fine.
Beginning with the field and vector space axioms might be fine for sophisticated students, but I don't think it would make for an optimal learning experience for most students. We also don't teach kids the Peano axioms before they learn to add and multiply
Re: The Little Book of Linear Algebra
#130Earlier quoted context omitted.
i don't agree because this seems circular. You cant even define a matrix as something that acts on vectors meaningfully until you have some machinery. if you start with a set S and then make it vector space V over field K. Then by definition, linear combinations (and its not an algebra so nonlinear isn't even defined) are closed in V. You can then define spanning sets and linear independence to get bases. From bases…
Where's the circularity? What you're saying is fine as an abstract presentation, but I was talking about how students might initially come to learn about matrices, so just introducing column vectors as representing points in 2 and 3 dimensional space and how matrices transform them is fine. Beginning with the field and vector space axioms might be fine for sophisticated students, but I don't think it would make for a…
But the question was about deriving the multiplication rule. I said you could derive it from systems of equations directly and gave a proof.
> and how matrices transform them is fine
this is circular. You are introducing/assuming the multiplication rule right here. You can't then derive it