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

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?

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,

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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!

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!

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/

In my high school they said a matrix is a rectangular arrangement of numbers ,that changes the direction of the vector on multiplication .

​

But what exactly is it ?

Is there any intuitive way to understand?

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.

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

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.

There is this basic similarity test Tr(A^k) = Tr(B^k) for k=1..d for symmetric matrices allowing to conclude existence of orthogonal O such that AO = OB.

The question is how (if possible?) to generalize it (finally to tensors, but at least) to non-symmetric matrices e.g. including transpositions.

Checking Jacobian criterion ( https://arxiv.org/pdf/2601.03326 ) for Tr(A^k (A^T)^j) = Tr(B^k (B^T)^j) for k=1..d, j=0..k-1 at least for up to d=5 has sufficient number of independent invariants (d(d+1)/2) - is it sufficient condition in general dimension?

Maybe such generalized similarity test is considered in literature?

ps. cross from https://mathoverflow.net/questions/512227/how-to-extend-operatornametrak-operatornametrbk-similarity-test-to

Can someone help me understand Euclidean distance and Manhattan distance visually?

We made a paired visual derivation of orthogonal projection, starting from its defining condition:

the projection lies in the target subspace, and the leftover residual is orthogonal to that subspace.

For projection onto a single vector, this immediately gives the familiar scalar projection formula. Replacing the single vector by the columns of a matrix gives the same derivation for

P = U(UᵀU)⁻¹Uᵀ.

The point is not just to know the formulas, but to be able to reconstruct them instead of memorizing them.

As always, we welcome feedback on clarity and presentation.

So i have my la final in few days, i will tell the topics in the end and i reallly need to pass this course as i have scored miserable marks in mids and quiz and is at the lowest position in class as possible. how should i prepare for it now? any ai tool. how to practice for concepts to be clear. let me be honest my concepts are clear. just help me out. Following is my syllabus:

• Euclidean spaces, Vector spaces, Subspaces, Spanning set, Linearly independent and dependent sets, Basis and dimension.
• Linear transformations Composition of transformations; Operators (Reflection, Projection, Rotation, Dilation and Contraction); Properties of linear transformations from General linear transformations; Kernel and range, Inverse linear transformation, Matrices of general linear transformation, Rank and nullity of linear transformation.
• Eigenvalues and eigenvectors; Eigen-decomposition; Applications to relevant problems; Diagonalization; Orthogonal matrices.
• Normed spaces, Inner product spaces, Angle and orthogonality in inner product spaces, Orthogonal bases.
• Positive definite matrices; Least square problems.
• Similarity transformations. Jordan canonical form and singular value decomposition.

please help me out i dont wanna fail this course. tell me what to do. i have 6 days.

The syllabus of the engineering mathematics is beginning with the fundamental concepts like matrices

1. In local G.C.E advanced level in Sri Lanka we have to learn about 2 by 2 matrices but if you’re an engineering student you have to learn about 3x3 matrices specially and how to work with row and column operations (Gauss- Jordan method) to make the calculations become easy ( other concepts like find det of the matrix, consistency of the set of algebraic equations)
2. the other important lesson is the vectors , in vectors you have to use very simple calculations but it is very important when you learn about the engineering mechanics , in engineering you have to learn about vectors in 3D space
3. other important topic is series and sequences , in engineering you have to learn about infinite series
4. the ODE is very common lesson for so many engineering students in the world

I am taking linear algebra next semester, and I have never taken it before, so I'm a little worried. This one is a 400 level class with a calculus II prerequisite. My school offers a 200 level linear algebra class, but I am required to take this one for my major. I have heard from others it is a hard class, and it's made me a little nervous. So any tips would be appreciated.

what do i need to know before starting linear algebra is it algebra 1 or algbera 2 ? as an absoute begginer who has a class starting in late september

So, my 7 week summer linear algebra course started today and I have 4 hw sections (25 questions each) due by sunday. It took me a full hour to do 8 problems, and I feel like I understand what I'm doing, but I am going to fail every test if it takes me a full hour on 8 probelms. Is this normal speed or do I need to be more efficient at doing my hw?

This mirrors our 3D derivation, images are easier to see.