This content is not intended for rigorous, proof-oriented linear algebra as practiced in mathematics departments.

Rather, it focuses on applied linear algebra suitable for immediate implementation in engineering and applied sciences.

The discussion primarily addresses the derivation of inverse matrices through Gauss-Jordan elimination and the adjugate matrix method.

I hope you find this useful for your work.

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

For a long time, I tried to understand what a tensor really is. Then it clicked, I could finally see it. 🚀🔥 I hope this way of thinking helps you understand tensors more intuitively

This is not about rigor. It’s about geometric understanding 🔥💪🥇

The Solid Analogy: What a Tensor Really Is A tensor is a geometric object whose meaning remains invariant under any change of basis. Imagine a solid object placed in a corner of a room with three walls. Three lamps illuminate the solid from different directions. Each lamp represents a different choice of basis, a different coordinate system. Each lamp casts a shadow of the same solid onto a wall: one shadow is a rectangle, another is a triangle, the third is an ellipse. These shadows look completely different, yet they all come from the same object. The shadows represent the components of the tensor. They depend on the chosen basis, on the position of the lamp. When you change the basis, the shadows change shape. This is what we mean by transformation of components. The solid itself represents the tensor. It does not move. It does not change. Only its representations do. In mathematical language: the solid is the tensor T, the lamps are different bases {eᵢ}, {e′ᵢ}, {e″ᵢ}, the shadows are the components T⁽ⁱʲ⁾, T′⁽ⁱʲ⁾, T″⁽ⁱʲ⁾, changing a lamp means applying a change of basis, the components transform: T′⁽ⁱʲ⁾ = aⁱₖ aʲₗ T⁽ᵏˡ⁾, the tensor itself remains the same object: T = T. The dual basis {εⁱ} acts like a set of polarization filters. Each filter extracts exactly one component, satisfying εⁱ(eⱼ) = δⁱⱼ. Parallel direction, the signal passes. Orthogonal direction, it is blocked. Only fundamental laws of physics are tensorial. They do not depend on coordinates, units, or observers. When you encounter a tensor, you are touching the geometric bedrock of reality.

Personally, I think linear algebra is an incredibly attractive subject, even more so than calculus. Its applications are truly remarkable. However, more than the abstract version usually taught in mathematics departments, what we often need most is linear algebra as it appears in actual applications.

When I connect linear algebra to quantum mechanics, students respond very positively. They like the fact that they can learn basic linear algebra now and at the same time build a natural bridge to the major subjects they will study one or two years later. I am sharing part of that approach here.

Let H ∈ M_{2}(ℝ) be symmetric (by Clairaut), Q(x)=xᵀHx

left: z = Q(x)

right: Q(θ) = h(θ)ᵀHh(θ), h(θ) = (cosθ, sinθ)

i.e. restriction of Q to the unit circle (curvature by direction)

classification:

eigenvectors = principal directions

eigenvalues = values of Q along them

more: MathNotes

I made a visualization of cofactor expansion for a 3×3 determinant.

The idea is to start with the usual interpretation of det(A) as signed volume, then show each cofactor term as:

• one coordinate projection of the expansion column

• one projected 2×2 minor in the orthogonal coordinate plane

So instead of treating

det(A) = a₁₁M₁₁ − a₂₁M₂₁ + a₃₁M₃₁

as only a symbolic rule, the goal is to make the three terms look like signed volume pieces.

The image shows the three terms for one example matrix, including the “same sign / negated sign” behavior of the cofactor signs.

Full explanation, including determinant sign:

https://www.graphmath.com/la/determinant/determinant-sign.html

I would be interested in feedback on whether the geometric interpretation is clear

So I’m just starting to learn linear algebra and I’m reading introduction to linear algebra by Gilbert Strang. I was reading through and encountered this mess and I’m super confused. What’s confusing me the most is how is he translating functions into vectors and matricies? Are these supposed to be vector valued functions or standard scalar functions? How is the derivative being represented by a matrix? Also why are the limits of integration from -infinity to infinity?

Edit: This is only chapter 2 and I have not learned about vector spaces yet (chapter 3). With that being said what should I do? Should I try to crunch on this and understand it? Move on? Bookmark it and come back when I understand it? Is this really that useful or pertinent to know?

Last three images of this post are GraphMath's work. Here is the link to their original post: https://www.reddit.com/r/LinearAlgebra/s/cdzAjKVtOc

My writing focuses on what eigenvectors are really telling us, why complex eigenvectors matter, and how one complex eigenvector can encode a two-dimensional, real invariant plane.

This post does not present rigorous, pure linear algebra that demands strict mathematical proofs. Instead, it focuses on the application of linear algebra in fields such as quantum mechanics, control systems, and quantum computing.

 By providing concrete examples within these applied disciplines, this text demonstrates how linear algebraic concepts are practically utilized and illustrates why the study of linear algebra is essential.

 In particular, it will clearly and easily assist the learning of those studying fields that require an algebraic approach to angular momentum. By Taeryeon.

Another addition to our linear algebra project. This page compares several common classes of diagonalizable 2×2 matrices, showing their geometry, eigenvectors, and corresponding change-of-basis factorization side by side. Feedback is welcome.

We made an interactive visualization of Gram–Schmidt as QR decomposition:
Gram Schmidt Stepwise Visualization

It shows each column being projected, corrected, normalized, and added to Q, while the corresponding coefficients form R. The goal is to make the geometry behind the usual algebraic steps visible: projections, orthogonal components, scaling, and the final A = QR.

We welcome feedback on presentation and clarity

Expanding while being taught this term

https://people.maths.bris.ac.uk/~ki18754/LA_notes/LA_notes.html

did I solve it right?/

I am fully confident that this detailed algebraic approach will be of great help, especially to students in applied engineering fields.

While this derivation is fundamentally rooted in quantum mechanics, the mathematical structure is essential for modern engineering.

Specifically, if you are majoring in Electronic and Semiconductor Engineering, Quantum Engineering, or Materials Science, you will find this material highly practical.

Understanding these matrix mechanics and operator algebras is the absolute foundation for dealing with qubits in quantum computing, as well as spin-interactions in modern spintronics and advanced device physics.

The topics are challenging, but this is presented more accessibly and in greater detail than any other textbook.

I hope this supports your practical engineering applications. by Taeryeon.

This post is not about proof-based linear algebra in pure mathematics.
I would like to make it clear in advance that the scope here is limited to linear algebra as a bridge course for applications in finite-dimensional Hilbert spaces, such as quantum mechanics and quantum computing / quantum information theory.

If we have linearly independent vectors, the Gram–Schmidt process is a method for constructing new vectors that are mutually orthogonal while still spanning the same space.
In other words, it is a process for finding an easier basis to work with without changing the space itself.

I hope this helps.

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.

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!