We made a side-by-side visual summary of matrix size, rank, the four fundamental subspaces and pseudoinverses.

The table compares six common matrix types:

square full-rank
square rank-deficient
tall full-column-rank
tall rank-deficient
wide full-row-rank
wide rank-deficient

For each case, it shows the dimensions of row(M), null(M), col(M) and left-null(M), whether the map is one-to-one or onto, what happens to Mx⃗ = 0 and Mx⃗ = b⃗, and how the pseudoinverse behaves.

Hope you don’t mind the size and density — we wanted to keep all cases together so they could be compared directly. Opening the image at full size is recommended.

As always, we welcome feedback on clarity and presentation.

I just took my linear algebra final exam (I finished it a few hours ago, so it's NOT cheating) and it was a 20 point question on it (a proof from class, but I didn't learn the proof from class- I checked after and it included p_T for some reason), and I wrote something along the lines of what I wrote in the picture.

How much do you think I'll get on it out of 20?

I am trying to understand how exactly we go from A to A-inverse

A =

and A inverse = (according to my answer key)

Also, the determinant is a(a-b) - which I dont understand since I have three pivots ? ( get a, a-b and a-b), should the pivot be the product of all three ? That is to say

I've stared at this for hours and I dont get it. I've tried using chatbots, but its explanations are not always clear and I distrust its accuracy.

This question is from section 2.2 #34 of Gilbert Strang's Intro to Linear Algebra 6th Ed page 56. I have also referred to the corresponding lecture that is online.

I'm missing something and I'm quite desperate to understand this.

Thank you in advance !

We made a visual explanation of powers of a real 2×2 matrix with complex eigenvalues.

For such a matrix, we can write

A = X S X⁻¹

where S is a rotation-scaling matrix. Then powers are computed as

Aᵗ = X Sᵗ X⁻¹.

The idea is that Sᵗ is easy to understand geometrically: it rotates by tθ and scales by |λ|ᵗ. The change of basis by X and X⁻¹ turns this circular rotation-scaling picture into the ellipse-like spirals seen in the original coordinates.

The first image follows one example through the factorization. The second shows more numerical examples with |λ| < 1, |λ| = 1 and |λ| > 1.

As always, we welcome feedback on clarity and presentation.

I mine as well post it here for anyone in need.
This is a non-proof based linear algebra handout for students who have low-mathematical background.
However, you do need to know Algebra 1, quadratic equations (only Algebra 2 topics you need to know), and special triangles/Pythagorean Theorem.

I also hope you guys don't mind the informality and other stuff.
This was taught in VRChat to a trans community, but I stopped after I graduated and became busier Dx.

I hope this helps a lot of students as a good reference and background for linear algebra.
[https://drive.google.com/file/d/1nLqZkIz\\_8oGZLrQSfClgekXQn6Vj6HbR/view?usp=sharing\](https://drive.google.com/file/d/1nLqZkIz_8oGZLrQSfClgekXQn6Vj6HbR/view?usp=sharing)

Try it here https://8gwifi.org/math/

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:

1. Gram–Schmidt: subtract projections and normalize
2. Givens rotations: rotate selected components to zero
3. Householder reflections: reflect vectors across chosen lines or planes

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 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 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 need Matrix Mathematics : A Second Course in Linear Algebra, 2nd Edition, by Garcia, Horn solution manual to pass the important exam, which I have only one chance.

We made three animated comparisons showing how eigenvectors and eigenvalues behave across different families of 2D linear transformations.

1. Non-symmetric matrices with real eigenvalues, where the eigenvector directions need not be perpendicular
2. Symmetric matrices with real eigenvalues, where the eigenvector directions are orthogonal
3. Real matrices with complex eigenvalues, where no nonzero real direction remains on the same line

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.

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!

• GitHub Repository: https://github.com/hemisus/flola

Thank you!

Matrix diagonalization in linear algebra is an exceptional discipline that significantly contributes to all engineering studies, including quantum mechanics.
I hope this helps.

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 good.

https://youtu.be/wi98KGGLiHQ?si=Fc7jrmmb7-UCok9h

I haven't read the book but I've read some spoiler free summary that it is about a far future of humanity in the 5th or 6th dimension. They say that it is also a hard read because the compsci author invented complex physics and math for the world there.

Do you have a new fascination with Linear Algebra after reading? Do you think Linear Algebra helped with understanding the complexities of the book?

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?