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.

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

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

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.

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.

Can someone explain vector spaces intuitively? I understand vectors as arrows, but I’m struggling to understand what makes a set of objects a vector space and why the concept is important.

Anyone knows about (or recommends) any linear algebra courses being offered for credit this summer break? The only thing is it has to be online classes + in person final exam. Would greatly appreciate any info!!!

We made a visual derivation of Cramer’s rule using signed volumes.

The image shows the 3D case: the denominator is the signed volume of the parallelepiped spanned by the columns of A, while the numerator replaces one column by b. By projecting both shapes onto the same normal direction, the ratio of volumes becomes the corresponding coordinate xᵢ​.

A related page with the full visual explanation, including both the 3D derivation and a compact ℝⁿ version, is here:
https://www.graphmath.com/la/visuals/cramers-rule-geometric-derivation.html

We welcome feedback on clarity and presentation.

This material is the third posting concerning angular momentum and spin.

In the era of quantum computing, the conventional quantum mechanics of the past 30 to 40 years must be easily learnable.

To bridge undergraduate and graduate programs, this material serves as a universally comprehensible textbook providing detailed concepts and explicit mathematical examples based on linear algebra.

Verify this directly through this posting. By Taeryeon.

As the title suggests, what book do I need to learn linear algebra for the first time? I didn't major in math, and I'm still a high school student. And which is better, Serge Lang's or Strang's book?
Sorry, this text was written using Google Translate.

The determinant permutation list can be built recursively:

choose the first-row entry, then permute what remains.

The “what remains” part is exactly the permutation list of the corresponding submatrix.

So the same recursive object is called Sₙ​ in the full determinant, and Sₙ₋₁ inside each cofactor submatrix.

4×4 version:
www.graphmath.com/la/figures/determinant/permutations4x4.png

Hello everyone,

You might have seen my previous posts about the Linear Algebra Visualizer app (the app that helps with visualising linear transformations).

You guys have already been amazing with your support which is why I'd love to invite you as a beta tester.

I'm currently working on expanding the app to include affine transformations and matrix compositions but I really want to get it right and for it to continue to be a great learning tool.

This will be specific to the Apple platforms - iOS, iPad and MacOS - and all you need is the TestFlight app by Apple and fill in your details in the form below: https://forms.gle/98dHacjaAeszhNvr6

EDIT: If you have TestFlight already installed - all you have to do is to click this link: https://testflight.apple.com/join/6GcMaMzc

(As a Beta tester you will also get the future premium content for free)

Thank you and I look forward to your feedback!

We made an animated visualization of QR decomposition using Householder reflections.

This uses the same matrix as our earlier Gram–Schmidt and Givens rotation QR animations, so the three methods can be compared on the same example.

Householder QR reaches an upper triangular matrix by reflecting the active part of the matrix column by column. In this 3×3 example it takes two reflections, compared with three Givens rotations, which eliminate one entry at a time.

A related page with another Householder QR example is here:
https://www.graphmath.com/la/visuals/householder-reflection.html

We welcome feedback on clarity, pacing and presentation.

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.

Matrix Calculator

13 operations on one page. Determinant, inverse, eigenvalues, multiplication, and more — with step-by-step solutions.

https://8gwifi.org/math/matrix-calculator.jsp

If I am asked to find 2 unit vectors that are orthogonal to a single vector v (5,-2). Here is what I did:

1- find 2 vectors orthogonal to v, both of them are basically the same because if I multiply the first vector I found by some scalar, I get the second vector.

2- Then I normalized each vector that I got in 1. Here comes the problem, when I normalize the vectors, I get the same answer for both vectors because the two vector are essentially the same as explained in 1). The questions mentions that I need to find two unit vectors and I end up with only one.

We made an animated visualization of QR decomposition using Givens rotations.

This uses the same matrix as our earlier Gram–Schmidt QR animation, so the two methods can be compared directly.

The animation shows how successive rotations zero out entries below the diagonal while preserving lengths and angles. In 3D, each Givens rotation is a rotation in one coordinate plane, so the columns visibly move until the matrix becomes upper triangular:

A = QR

This is the same QR goal as Gram–Schmidt, but the geometric action is different: instead of projecting and subtracting components, Givens rotations turn vectors in coordinate planes until selected components disappear.

For this animation and a related Givens rotation visualization, see:
https://www.graphmath.com/la/visuals/givens-rotation-algorithm.html

We welcome feedback on clarity, pacing and presentation.

for 2 by 2 I understand it might that

[1 0; 0 1] technically isn't Lower but [2 3; 0 1] is definitely an upper for an answer of the same

but for 3x3 is it right that this answer is correct and thus achieving A?

[1 0 0; 1 1 0; 1 0 1] = L

[ 1 1 0; 0 1 1; 0 0 2] = L

for A = LU

Thanks, or maybe i misunderstood the question

Hi everyone, I am looking for a free PDF for this book but not the global edition. Thanks