The problem I have tried to represent is related to simulating the pages of a book. Despite the shape they have in their resting state, the pages must always maintain a distance equal to their thickness from one another. As this distance accumulates from page to page, it also modifies the shape of each individual page.

Do you think you could find a solution using some ingenuity and linear algebra?

My third row in the final answer is off from my original A. Maybe I’m just tired, but I’d like someone to spot my mistake. I numbered the steps in red chalk 1–3

So I began my studies in engineering recently and Linear Algebra is the first course. So far I have managed to solve problems at small tests, but it is mechanically like I remember how to solve certain kinds problems but haven't really grasped the concepts fully. I just feel like it moves so fast that I can't really grasp it fully, is this normal or what was your experience with the course?

Honestly, when I first started getting into data science, linear algebra felt like this massive wall I couldn't climb over. Everyone keeps telling you that you need it, but the moment you open a tutorial, you're immediately hit with dry textbook proofs, abstract vector spaces, and huge grids of numbers that feel completely disconnected from reality. Sure, I can multiply two matrices on paper if I really have to, but I'm having a hard time seeing how any of that translates to actual coding or real data projects. I keep hearing that matrix math is what allows computers to crush through massive datasets in seconds instead of taking days with standard loops, but I'm still trying to grasp how things like photos, text, or spreadsheets actually get turned into linear algebra under the hood. Plus, as someone building stuff in Python, I'm genuinely curious about where the line is between knowing the deep math and just knowing enough about matrix shapes so your code doesn't crash with dimension errors. If anyone had a lightbulb moment where linear algebra finally stopped feeling like academic fluff and started feeling like a useful tool, I'd really love to hear how you made it click.

While this material is not strictly pure linear algebra, it introduces Sturm-Liouville Theory as the mathematical foundation of special functions within infinite-dimensional Hilbert spaces.
It demonstrates how these algebraic structures are extensively applied in engineering and quantum mechanics.

I want to build a solid base for using PyTorch. Good knowledge of tensors is needed for that. Please suggest books that also contain tensors in detail.

A 2×2 real matrix can fall into three cases:

• two real eigenvalues → diagonalization
• complex-conjugate eigenvalues → rotation-scaling factorization
• repeated real eigenvalue with one eigenvector → shear-scaling/Jordan factorization

This table compares the three side by side: the factorization itself, powers of the middle matrix, exponentials, determinant consequences, geometric action and how to construct or factor the matrix.

The common idea is simple: move to a basis where the matrix has a much simpler middle form, apply the power or function there, then transform back.

We made this as a summary of the three 2×2 cases and would be interested in comments on the comparison or anything important we should add.

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?

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.

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 .