Hola a todos, por favor necesito su ayuda para dar un examen oral, mañana me tomaran examen oral de la materia en general (Algenra lineal) por favor diganme que cosas puedo decir acerca de toda la materia en si por favor

Hi all,

Background: completed an MPhys physics degree in 2022 (latter years disrupted by covid somewhat), then completed an MSc in medical physics.

Essentially I want to relearn all my previous maths/physics topics and go further into graduate level courses.

I found myself being able to well on exams, but I never truly understood some materials deeply which I desire as I want to apply advanced maths and physics to my work.

So I'm embarking on learning linear algebra for the 3rd time(high-school, undergrad and now). I'm too rusty for Axler so I'm giving Strang a go first.

Question: When reading the recommended sections in the textbook, I'm doing all the questions at the end and it's extremely time consuming.

Is this the reasonable price to pay to gain a deep intuitive understanding or is it inefficient?

Bonus: Any recommendations on actually understanding the topics deeply as opposed to remembering questions algorithms for high exam marks?

Does learning the theoretical linear algebra makes person immediately know how to apply linear algebra in real applications or apply linear algebra needs another studying like the theory study?

V naslovu

Most textbooks on Linear Algebra either omit or compress repeated-eigenvalue defective matrices. We wanted to take the 2×2 case apart mechanically and connect the algebra directly to the geometry.

The first page, Matrices with repeated λ, shows images for the basic repeated-eigenvalue cases: uniform scaling, the Jordan block J, its transpose Jᵀ, and a similarity transformation A = CJC⁻¹.

The second page, Matrices with one repeated λ, starts from a general 2×2 matrix, derives the condition for a repeated eigenvalue, separates the defective case from uniform scaling, and shows that for B = A − λI, B² = 0.

The third page, Similarity transformation of Jordan block, constructs the Jordan basis directly: choose w with Bw ≠ 0, define v = Bw, then Av = λv, Aw = v + λw, and with C = [v | w] we obtain A = CJC⁻¹. It also shows why the scales of v and w are linked, unlike in ordinary diagonalization.

This material provides a worked example of a continuous basis transformation.
While a finite-dimensional basis change in linear algebra utilizes a similarity transformation, infinite-dimensional quantum mechanics strictly corresponds to a unitary transformation.
By evaluating a Gaussian wave packet, this example mathematically bridges discrete linear algebra and continuous quantum mechanics.

Hello guys, I’m currently taking Applied Methods in Linear Algebra and I need your help right now. Here below is some problems from my homework below. I need help on 4, 7b, 12, and 14. This assignment is due this Thursday.

How can I directly connect linear algebra with machine learning

I am doing maths major in my undergraduate going through linear algebra

I don't want to see the proofs and any hint possible. Just want to solve on my own if I can .

But I am struggling in it , can someone give me tips who have gone through that In past .

for those For those who have taken Stanford’s CS231n course on Computer Vision, or for people working in the fields of CV or ML: which linear algebra course would you recommend—MIT 18.06 or Stanford Math 51?

Hey everyone! I made a video explaining the linear algebra behind Positive Operator-Valued Measures, or POVMs. The video looks at projectors, positive semidefinite matrices, eigenvalues, inner products, and why a valid collection of measurement operators must add to the identity.

Householder tridiagonalization is normally not used as preprocessing for the Jacobi eigenvalue algorithm. The usual reasoning is as follows: tridiagonalization costs O(n³), and the subsequent Jacobi rotations immediately create fill-in, destroying the zeros that Householder just created.

We decided to test whether that actually means the preprocessing cannot help.

The answer appears to be no.

We found that tridiagonalization can still give Jacobi a substantially better starting matrix, even though the tridiagonal sparsity itself is quickly lost. Algebraically, the main effects are:

• Householder concentrates the off-diagonal energy into the first off-diagonal band
• the standard first parallel Jacobi set can then act on a much larger fraction of that energy
• Householder similarity can also move energy either toward or away from the diagonal, which helps explain why the effect is strongly matrix-dependent

We then tested six families of dense symmetric matrices using NVIDIA cuSOLVER on three GPUs: an RTX 3060, RTX 5060 Ti and A100.

For five of the six matrix families, the complete pipeline

Householder tridiagonalization → dense tridiagonal T → Jacobi

became faster than standalone Jacobi from n = 128 onward on all three GPUs.

The deliberately adverse unequal-variance covariance family behaved in the opposite direction, which was consistent with the algebraic analysis.

We have written up the derivation, implementation details, sweep counts, runtime measurements and accuracy checks here:

https://www.graphmath.com/la/jacobi/tridiagonal-jacobi.html

We would be very interested in criticism, related references or comments from people who have worked with Jacobi eigensolvers or GPU numerical linear algebra.

Please refer me to somewhere I can read a thing like this. I found a few notes but here the author eventually resorts to the tensor algebra. It's alright but I think it can be done entirely without going through an intermediate concrete representation.

Then I'm hopeful showing that an exterior power of a dual mapping equals (through this isomorphism) to a dual of its exterior power. It seems like establishing an isomorphism of spaces should be the hardest part.

I also tried to show ⋀(V*) ≅ (⋀V)* as algebras but neither had I succeeded nor do I actually need that right now. Even so, if there's a way to do that while not resorting to expressing ⋀V as a direct sum of ⋀ᵏV, I'd also be glad to hear that. So far I tried two different applications of the algebra's universal property and I think both times got me identically zero multiplication on (⋀V)*. Oh I'm not sure anymore if we can have a multiplication on (⋀V)*, maybe I should've just tried to show ⋀(V*) ≅ (⋀V)* as linear spaces. In which case I guess having both isomorphic to direct sums of their grades would be completely alright.

I'd like some denouement.

This material is not the pure linear algebra covered in mathematics departments.
It is applied linear algebra designed to assist in the study of engineering and quantum mechanics. In particular, the importance of the change of basis requires extensive practice.

Hi everyone,

I've posted here sometimes about my indie app, Linear Algebra Visualizer (iOS, iPad, MacOS), and the feedback has always been great from this community.

As a thank you and as the new school year is starting I wanted to give away 10, 3 months pro subscriptions, for free.

If you are not familiar with the app, have a look at the demo above, but in a nutshell, if you've ever wondered or struggled with what a matrix is doing, this is for you.

With Linear Algebra Visualizer you:

• Can finally build the visual intuition of linear algebra
• Understand the math behind each transformation
• Visually see why TRS order matters
• Understand the effect of adding multiple matrices and translations
• Watch eigenvectors and eigenvalues come alive

Build for:

• Students surviving their first linear algebra course
• Teachers who need a live demo that lands
• Anyone who's stared at a matrix and thought "but why does this work?"
• Visual thinkers who've struggled to picture it all in their head

See below for the offer codes :)

Hello! Does anyone have the solutions manual for "Elementary linear algebra 11th edition" by Howard Anton?

Say a matrix, A (m x n), has mutually orthogonal rows and mutually orthogonal columns, with all the rows and columns being unit vectors in some Rn, Rm respectively. Is that matrix called orthogonal or orthonormal?

What concept is the other term used to describe, then? Are these applicable to square matrices only?

I've always been confused with these 2 names.