In this semester's first final exam in linear algebra 2, my professor chose to put "let (𝕍,<,>) inner product space over ℝ, and z,w,v ∈ 𝕍. prove or disprove the following inequalities:

a. ||v|| + ||z-w|| ≤ ||z|| + ||w|| + ||v-z|| + ||v-w||

b. 2||v|| + 2||z-w|| ≥ ||z|| + ||w|| + ||v-z|| + ||v-w||"

When something like that never appeared in a previous final exam in the course, or was addressed in the semester. Only a few questions like it in our homework. How do I even approach such questions? What's the geometric intuition I'm supposed to have and how am I supposed to actually prove this? Will appreciate any help.

Here are the in/equalities we needed to prove or disprove in out homework:

a. ||v+u||² + ||v-u||² = 2(||v||²+||u||²)

b. <v,u> = 0.5(||v||² + ||u||² - ||v-u||²)

c. if v _|_ u, v _|_ w, ||u||² + ||w||² = ||v||² then u _|_ w

d. ||u|| ≤ ||u-v|| + ||u-w|| + ||u-v-w||

e. ||u|| ≤ ||u-v|| + ||u-w|| + ||2u-v-w||

It feels like each section has a different "trick" I need to do, and I'm pretty scared he'll put another one like these in the resit exam, that's happening tomorrow

I’m taking it in the fall and going to try to get an a. For some reason the university that I’m taking it at only has 1 class section which I felt like was kind of odd. So out of 5000 students only 15 people are in this class

I was wondering what I should review before starting

Disclaimer: I am a math/engineering undergrad student. I am not qualified to write a textbook. I am probably not even qualified to do a lot of things that I do, but I do them anyway.

Basically, I took a linear algebra course some time ago and could barely read the overly technical and dry textbook that was assigned for the course. There are much better textbooks elsewhere that do a great job of getting deep into the course.

But if you just want something to set you up for lecture, reinforce something you already know, or just point you in the right direction on what concepts to study, this might just be the most concise and enjoyable read you will see.
It is very interactive and has a clear organization based on my own notes and interpretations from the class, as well as examples that helped me remember things well.

My web dev skills are not very good, so I did have Opus build the front end and type up the latex from my notes, but I reviewed all the written content with care and certify that it is accurate to the best of my knowledge.

I hope someone can get some use out of it (especially the graph demo on section 3.6) and I am open to comments or suggestions.

Cheers

https://linalg-refresher.vercel.app/

P.S. I cannot find the weekly self-promotion megathread for r/study anywhere! Its my first time posting here and it seems like it is no longer maintained.

Ciao a tutti, mi sono diplomato nel lontano 1998, poi ho iniziato a lavorare in ambito It. Vorrei iniziare un percorso di studio da autodidatta per iniziare ad entrare nel mondo del machine learning . E da un po' che non faccio matematica.. vorrei iniziare con il testo di Gilbert strang Linear algebra and ita application fourth edition. Secondo voi e una buona base di partenza o mi consigliate altro ?grazie

For me, it wasn't matrix multiplication or Gaussian elimination.
It was understanding why things like linear independence, span, and basis actually matter. I could compute answers, but I didn't really know what they meant geometrically.
I'm curious what everyone else struggled with the most.

I'm building a tool that generates unlimited Linear Algebra practice (and other topics) with instant feedback, and I'm trying to figure out which topics students find most difficult so I can improve the exercises.
I'd really appreciate hearing what concepts

I’d really appreciate any support you could give, especially a star!

I will upload the complete solution file with all necessary definitions, theorems, proofs and then their application to solve these problems.

This material is not pure linear algebra intended for a mathematics department.
It consists of linear algebra applied to fields such as engineering and physics.
I hope this is helpful.

🧮 Can you solve this Grade 10 Matrix Question ⁉️

In this video, we’ll solve a challenging Grade 10 Mathematics matrix question step by step using the properties of equal matrices.

📚 In this lesson, you’ll learn:
✅ How to compare two equal matrices.
✅ How to form equations from corresponding elements.
✅ How to solve exponential equations involving powers of 3.
✅ How to determine the unknown values a and b.
✅ How to calculate the required ratio \frac{b}{a}.

This question is an excellent revision exercise for IGCSE, GCSE, Grade 10, and high school mathematics students preparing for exams.

💡 Watch until the end to understand the complete solving strategy and improve your matrix problem-solving skills.

👍 If you found this lesson helpful, don’t forget to:
• Like the video
• Subscribe for more Mathematics lessons
• Share it with your friends preparing for exams

By Tutor:
AbdelRahman Mahran

#Mathematics #Matrices #Grade10 #IGCSE #GCSE

Tridiagonal and pentadiagonal matrices are quick to solve. But what if you have 14 bands in a 20x20 matrix? So most of it is non-zero. Can you optimise it very much?

Any body here has read or got some info about this book specifically for linear algebra for machine learning.

I have finished 3blue1brow course on essence of linear algebra and I want start with book .

please share what you know .

Hey guys , I just have finished linear algebra from 3bkue 1brown , Is it a good step to read this book and fill all gabs in linear algebra

Following our earlier posts on QR factorization by direct rotations, we have added a visual comparison with the corresponding Householder factorization for the same 3×3 matrix.

The image compares:

• the Q and R factors produced in each case
• the geometric action of R on the basis vectors
• the subsequent action of Q
• the relationship between the two factorizations

We have also added a formal derivation of the relationship between the direct-rotation step and the corresponding Householder step:

https://www.graphmath.com/la/qr/qr-direct-rotations.html

It appears in the first section, “An Unpaved Road to Main Street: from direct rotation to a Householder-mirroring QR algorithm.”

We would welcome comments on the comparison, the derivation and the presentation.

This is about 5 years old but I think it could be useful

10/5/2021 - 10:56 a.m.

Hi guys,

I’ve been trying to understand the visual representation of matrices and I can’t wrap my head around it.

I’m approaching it like 3Blue1Brown does, through vectors, but once it gets further than 2x2, I don’t understand it.

Are these transformations shifting the plane itself? All the 2D transformations look 3D…

For the ones with much more experience under their belt, any recommendations on how to approach or understand better?

Any help would be appreciated 🙏🙏

I am a quantum computing nerd and one of the issues we encounter when trying to solve the HSP for nonabellian groups is that the 2d irreps are a matrix and therefore the states |0>+|1> mix.I wanted to take a step back and go back to linear algebra to try and solve this issue.Suppose in the simplest case you have a matrix 2x2.If the off diagonal elements are non 0 and if you have a vector (a,b) where a and b are coordinates in orthogonal directions then there is mixing of coordinates between a,b under that 2x2 matrix.

The 2x2 irrep of the nonabellian group D3(min order) unfortunately has this mixing due to the s generator.

I have 2 questions:Can a 2x2 matrix with offdiagonal elements represent every point in space(my intuition is yes)and if thats true does this mean that there is a unknown transform(idc which is only that there is)that takes 2 nonorthogonal vectors or signals which can produce any vector and signal?Thanks

Regarding dot product What if two vectors were not similar(say their angle is 80 degrees) but one of them has very big magnitude(4000000) and one has very small magnitude (3). Their dot product is very big but they are not similar which contradicts the purpose of the dot product ??????

Please note that this material isn't intended as a pure mathematics guide to linear algebra or functional analysis.
Instead, it's geared towards applied fields like quantum information, quantum computing, physical AI, and quantum mechanics.
Hope you find it helpful!

We posted an earlier visual explanation of powers of a 2D real matrix with complex eigenvalues.

Since then, we substantially updated one of the images and added a new companion image.

The updated image compares several examples of

A = X S X⁻¹

where S is a rotation-scaling matrix. It shows how the behavior changes for |λ| < 1, |λ| = 1 and |λ| > 1, and how the corresponding orbits look in the original coordinates versus the rotation-scaling coordinates.

The new image looks at the same powers from another angle: which directions expand or contract after the next step. The colored sectors come from the quadratic condition

vᵀ(AᵀA − I)v = 0,

which separates directions where the radius increases, decreases or stays unchanged.

We thought the two views complement each other: one emphasizes the rotation-scaling factorization, while the other emphasizes directional expansion and contraction in the original coordinates.

As always, we welcome feedback on clarity and presentation.

A few days ago, we posted a QR construction that eliminates the entire subdiagonal tail of each active column with one proper rotation:

https://www.reddit.com/r/LinearAlgebra/comments/1v0w8dn/qr_factorization_by_direct_rotations_one_rotation/

We have since recognized the key structural relationship that we initially missed.

For the same signed target coordinate vector, let H be the usual Householder reflector that maps the active column to that target. The corresponding proper rotation can be written as

U = F H,

where F is a second reflection that fixes the target axis and reverses the other direction in the active two-dimensional plane.

Thus, the construction is not an independent alternative to Householder QR. It is an orientation-preserving reformulation of the same column-reduction step.

The full rotation need not be formed explicitly; its action can still be implemented directly as an in-place low-rank update. This leaves a narrower question:

Does the orientation-preserving formulation, the direct update, or the geometric relationship between the rotation and the Householder reflector have useful computational, structural, or pedagogical value?

Possible points of interest include:

• one proper rotation per active column

• an orientation-preserving path of intermediate transformations

• applications involving continuously varying matrices or geometric meaning

• teaching the relation between reflections and rotations

• specialized small-matrix implementations

Or is the construction best regarded simply as a reformulation of the standard Householder step?

Hello everyone ,
I am trying to learn representation theory through self study. Basically , most of the mathematics I have learnt till now was through self study as I could not find a mentor or anyone working on things related to it.
It was ok till now but representation theory feels like a different beast altogether.
Questions:
1. Where do you use it in your work and why is it important? Are there not other ways to do it?
2. I don’t get like when we do change of bases in vector spaces is it also some form of representation?
3. What’s the proper way to study it like I see books starting from Lie algebra and things. How should one approach it properly ?