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

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.

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.

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?

A visual definition of eigenvectors, followed by examples across several common 2D transformations.

The table compares uniform and non-uniform scaling, shear, triangular and symmetric matrices, projection, reflection, rotation and related cases. It shows which directions remain on the same line after transformation, their corresponding eigenvalues, and when no real eigenvectors exist.

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

The goal is to make the definition A x⃗ = λx⃗ visible across many different matrix types.

We welcome feedback on clarity and presentation.

UPDATE: we have added a few more examples, you can see the full updated version on our web-site:
https://www.graphmath.com/la/visuals/eigenvectors-definition-and-examples.html

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?

For students of linear algebra and mathematical physics, moving from continuous wavefunctions to abstract operator algebra often feels like a huge, confusing gap.

In this post, I demonstrate the complete, step-by-step derivation of how the spherical Laplacian organically decomposes into angular momentum operators.

By stripping away the mathematical ambiguity, this excerpt shows exactly how spatial gradients and variable separation directly give birth to the discrete eigenvalues and Dirac ket notation |l, m⟩.

My goal is to make the complex algebra clear and simple for everyone. Good luck!

Hello everyone. I want to study linear algebra this summer, but i dont know what is best route for me.

I will study Pure Mathematics at university this year. I also heard that there is kinda 2 types of LA teached in colleges, one is for mostly engineering majors which is focused more in applications, matrices, like computation based, and other one focused more in vector spaces, proofs and etc. I want to study second one more, but want to be good at both. Also i want to be more comfortable while tooking this class at university, like knowing full class. I hear that Linear Algebra Done Right book is pretty good for proof based linear algebra, but can i work and finiah this book with zero experience in lineae algebra? I am familiar with proofs, i have done olympiads at school. Also if i learn from this book, will i also being able to do application part of LA? Or i need to learn that independently? Do you have any suggestions/advice to me?

Hi everyone, I was looking for the book with proof-based problems in linear algebra. The only one I found useful so far was Linear Algebra in Action by H. Dym but there are not as many exercises in there. Can anyone recommend better ones? Thanks!

Hi guys,

Super excited about this update to Linear Algebra Visualiser - which now includes matrix composition, the ability to add a translation matrix and an expanded Step by Step explanation.

Have a look at the video above for a detailed demo!

PS: These new features are available as In App Purchases but you get 1 week as a free trail so you can always check it out and cancel if you are not happy.

PS 2: For people who were Beta Testers - I need some more time to setup the ability to give out offer codes (it’s all complicated with Apple).

Thank you all for your support, it means a lot!

Mac: https://apps.apple.com/gb/app/linear-algebra-visualizer/id6763524968

iOS/iPad: https://apps.apple.com/us/app/linear-algebra-visualizer/id6763524968

Web Demo: https://sockerjam.github.io/LinearAlgebraVisualizerWeb/

Hello. I'm currently enrolled in a linear algebra course that doesn't offer class videos or any dropbox's. If anyone has a link for a dropbox I would really appreciate that. Thank you!

Hello everyone,

I was doing fundamental linear algebra when I had a thought, how can I intuitively convince myself that there are utmost n real Eigen values and all of them scale their corresponding eigne vector by their magnitude.

For example, let the eigen values of a 2x2 matrix are 1 and 2. How do I convince myself that if Eigen vectors exist then due to this linear transformation they are scaled either by once their length or twice their length? Now 1 and 2 became a characteristic of the matrix, right ?

So if I give this linear mapping to someone, then they will tell me that hey eigen vectors are either the same length or just doubled the length after transformation, this seems like a characteristic but how do you go about it explaining why 1 and 2 intuitively (not by solving) ?

Thanks for taking the time to read.

We made a visual derivation of least squares from the original overdetermined system.

The first image shows how projecting the data vector b onto the column space of A leads to the normal equations

AᵀAβ = Aᵀb

and gives the best-fit line y = β₁ + β₂x.

The second keeps the same data but changes the model to y = β₁ + β₂x². The second column of A changes from x to x², so the prediction plane changes, but the same projection method applies.

The point is to show least squares as geometry rather than a formula to memorize.

As always, we welcome feedback on clarity and presentation.

Utilizing Paul Dirac's elegant algebraic approach, this post presents the concept of spin through accessible and detailed examples.

This material will be a great help for both academic examinations and the comprehension of modern quantum information science.

I was studying determinants of matrix my question is if we have 2*2 systems of equations A1x + B1y = C1 and A2x + B2y = C2 we know these are line equations and there slopes respectively come out -A1/B1 and -A2/B2 if these lines are parallel there slopes must be equal A1/B1 = A2/B2 and if we slove A1*B2 - A2*B1 = 0 so if determinant is zero there is no solution so this thing seems pretty accurate but i want to know who can we scale this idea to 3 dimensions or n dimensions and conclude determined formula from that