I’m looking for a math tutor to help my daughter with Linear Algebra and Differential Equations. She is currently a sophomore at IVC. I would really appreciate any recommendations—thank you in advance!
Hey guys, I'm wondering which linear algebra course I should take. Has anyone taken these courses and can give me a review?
We previously posted separate 3D animations of QR decomposition by Gram–Schmidt, Givens rotations and Householder reflections.
Here is a 2D comparison of the same three geometric routes.
The goal is to show that all three methods reach the same kind of result, A = QR, but by very different geometric actions:
In 2D, the Givens case is almost trivial: only one rotation is needed to zero the lower-left entry. In higher dimensions, Givens QR proceeds by many such rotations, one entry at a time.
One detail worth noticing: Gram–Schmidt, in its standard form, produces positive diagonal entries in R. Givens and Householder versions may produce different signs depending on rotation/reflection sign choices. This is normal: QR is unique only after an extra sign convention is imposed, such as requiring positive diagonal entries in R.
More explanation and the 3D versions are here:
https://www.graphmath.com/la/visuals/qr/qr-three-geometric-routes.html
We will also add these 2D animations to that page.
We welcome feedback on clarity and presentation.
I already understand the algebraic proofs using the Fundamental Theorem of Linear Algebra, Rank-Nullity Theorem, Gaussian elimination, etc. Those are clear to me.
What I'm looking for is the deep intuition behind why this has to be true.
In other words, why is the dimension of the column space always equal to the dimension of the row space of the same matrix?
Geometrically, the column space and row space live in different vector spaces (R^m vs. R^n), so it isn't obvious to me why they should always have the same number of independent directions. What is the underlying constraint that forces this equality?
I'm not looking for another algebraic derivation. Instead, I'd love explanations that answer questions like:
What is the geometric picture?
Is there an information-theoretic, transformational, or degrees-of-freedom interpretation that makes this equality feel inevitable rather than something we simply prove algebraically?
Are there any visualizations or mental models that make this theorem "click"?
I'm especially interested in explanations that make the result feel almost obvious once you see the right perspective.
Edit:
I know most of the popular formal algebraic proofs to prove this, what i am looking for is intuitive perspective
For example, we can intuitively understand why
rank(A) + nullity(A) = n
When we apply the transformation A to vectors, each independent direction has only two possibilities: it either survives (maps to a nonzero independent direction) or it is killed (maps to the zero vector). Since these are the only two outcomes for the n independent input directions, it is intuitive that
rank(A) + nullity(A) = n
I'm looking for a similarly intuitive explanation for this theorem. Rather than an algebraic proof, I want a geometric or conceptual way to understand why it must be true
I think the title does not do much justice to my question but it would have been very long otherwise.
I have been trying to implement the TurboQuant Algorithm from google on my own. It is a simple rotation based KV cache optimization technique for transformer.
The base is that K which is a N dimensional vector usually has its components very high in some places and very low in most. They say that rotating by orthogonal matrix spreads the components of such vectors evenly while preserving inner products. I clearly understand this part.
The area that I dont understand is where they Quantize these components using centroids calculated by Lloyd Max Quantization(LMQ).
The basic Algorithm is that a quantization interval v_k = E[m / mk-1 <= m <= mk] where m are the quantization intervals and v_k is the kth quantization level. so it includes an integral over the Probability distribution of m.
The thing is, the paper chose to use a Gaussian Distribution for this F(m). Our goal is to minimize the quantization error. So to minimize it, we kind of also need the values of the Rotated vector to fall under a gaussian. But from all I understand, the hadamard transform just smears the value of peaks and converts the values to be quite uniform.
My conclusion was that the rotation somehow generates values that are a part of gaussian distribution. I just dont know if I am wrong or right.
I am sorry if my explanation is fundamentally wrong somewhere. Thank you!
I am an ex Microsoft Senior engineer. I have created this video explaining backpropagation using equations, deriving each equation by hand. Can I have some feedback? Thanks much
We made three animated comparisons showing how eigenvectors and eigenvalues behave across different families of 2D linear transformations.
In each animation, the transformation develops continuously from the identity matrix to the displayed matrix. The goal is to make the difference between these three cases visible rather than only algebraic.
Full-size animations and explanations:
https://www.graphmath.com/la/visuals/eigenvectors-eigenvalues-2d-transformations.html
As always, we welcome feedback on clarity and presentation.
How can I learn Linear Algebra effectively ? Should I go to AI tools or Watch videos of Gilbert Strang Sir . I am having trouble with this course. I can't change it or choose course according to my choice. I wanted some suggestions from those who struggled with this course at first and then discovered the effectively way of learning it. Please help me
Matrix diagonalization in linear algebra is an exceptional discipline that significantly contributes to all engineering studies, including quantum mechanics.
I hope this helps.
Hi everyone I'm struggling with the content and my method of study currently is going through the tutorial questions but I feel like it's not effective as I forget how to do it once i move onto a different section. Please share your insights on how to study and if anyone can mentor me through this course let me know thanks in advance.
Hi all, I have been now just writing things to consolidate some basic applied math. This is nothing advanced but just a good way to put things out and to learn by writing.
One of the things I try to do is to build things more intuitively instead of the traditional math book approach of starting from the final formalized result and giving a few formalized hints on how that object came from.
A bizarre linear system for neural networks applications:
https://archive.org/details/the-oddness-of-hd
One softwarez is:
https://archive.org/details/hd-dh-hd-dh-graph-viewer which we looked at before.
Another softwarez is: https://archive.org/details/h-12-d-dynamics
but that is for the atypical H₁₂ system, but you can still see the oscillations in some configurations. If you switch configurations you can see transitory oscillations as well. That is not due to dissipative effect, it is due to energy only slowly draining out of the old oscillation modes into the new one.
I keep getting banned on physics and neural network forums about this, where it might more properly be discussed. They really are incapable of absorbing information from "orthogonal" channels.
Hello.
I recently built a Linear algebra editor for a university assignment (JAVA).
It supports basic operations (arithmetic, matrix decomposition, and SVD).
And since my original goal was to visually represent AI model architectures, I also added operations like Convolution and Reshaping (View).
I'm not entirely sure how practical it will be, but I wanted to share hoping it could be helpful to some learners who are trying to grasp these concepts.
Also I'm not an expert in linear algebra myself...so if you notice any issues, or have suggestions, please let me know. Any feedback is appreciated!
Thank you!
Hello,
I am in my second year of university doing a life science degree. I hope to specialize in biophysics someday. I took Linear Algebra I instead of Calculus II in first year to fulfill a math credit, but ended up really liking it.
But since Calc II is a prerequisite for Lin Alg II, I cannot continue taking this class, but I really do enjoy lin alg and find it fascinating. Taking calc is not an option for me at the moment as I did absolutely horrible in calc I and I know that will not go well.
Are there any fields or specialties that combine linear algebra and sciences? And if I were to self-study, what would be the best order to approach topics? My course was more computation than proofs based, so I'm a bit nervous about getting into that.
Edit: I know I likely won't get very far, I just think it'd be a side quest I'd do for fun
I appreciate any guidance, many thanks :)
So I planned on taking two courses over the summer to make sure I stay on track for my major. Both are 5-week courses; the first one started earlier this week, and the second one starts at the end of July.
Honestly, the first week has not been fun. While I understand the basic concepts and how to solve the problems, I am already very far behind. I really don’t think I’ll be able to thoroughly prepare myself for the midterm, which is this coming Wednesday.
I usually need a bit of time to fully digest new concepts, and this class seems to require a very different skill set compared to something like Calculus.
I’m seriously considering dropping it, but I’m on the fence. Would anyone recommend dropping this course now and taking it through a community college (CC) during the fall quarter instead?
Hi guys,
I created a youtube video explaining how eigenvectors work, its applications including many visual elements and animations.
The video turned out to be a bit long (40+ minutes), but I was personally quite happy with the content itself. Would appreciate your feedback on whether this video is helpful.
I’ve been thinking about the sample mean from a linear algebra perspective.
If y is a data vector and 1 is the vector of all ones, then the average can be seen as the scalar you get when projecting y onto span(1).
So the projection has the form:
y-hat = y-bar · 1
where y-bar is the usual sample average.
I like this because it makes the average feel like the simplest possible least-squares problem: find the constant vector closest to the data vector.
It also connects naturally to ordinary least squares regression, where y gets projected onto the column space of X instead of just the one-dimensional space spanned by 1.
Does this seem like a good way to introduce projections/least squares, or would you teach it differently?
Random post but are there are any applications of these invariant subspace decompositions outside pure math? There was a NASA paper on control theory which involved cyclic decomposition but that was from decades ago. Any modern applications? For example ML related?
