Loading video...

Video Failed to Load

There was a problem loading this video. This could be due to a temporary network issue or the video might be unavailable.

the first free interactive linear algebra textbook: - vectors and matrices - dot product and vector product - determinants - matrix ranks - eigenvalues and eigenvectors

ℏεsam

72,907 subscribers

63,641 views • 1 year ago •via X (Twitter)

Science & Technology Education Health & Wellness

Anya Rossi• Live Now

Private livecam show

4 Comments

ℏεsam1 year ago

🔗

Sayan Chakraborty1 year ago

damn, where was this when I was growing up. beautiful!

steve ike1 year ago

Love the idea of interactive textbooks.

Aditya Dixit1 year ago

Interactive math textbooks are the way to go! Thanks for sharing.

Related Videos

🧵1/n This thread covers the linear algebra for modern computer graphics, bridging geometry, physics, and computation. Video 1: Vector Spaces & Cartesian Coordinates Source: CMU. 👇Video 2: Vector Operations, Vector Spaces, Functions as Vectors, and Vectors in Coordinates

🧵1/n This thread covers the linear algebra for modern computer graphics, bridging geometry, physics, and computation. Video 1: Vector Spaces & Cartesian Coordinates Source: CMU. 👇Video 2: Vector Operations, Vector Spaces, Functions as Vectors, and Vectors in Coordinates

tetsuo

52,043 views • 1 year ago

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

The Trap in Every Mathematics Lecture If you’ve taken enough math courses, you start noticing the same little move. The lecturer warms up with the obvious stuff, add matrices entrywise, scale by α, do the row-column product, and you’re thinking alright, where is this going. Then you relax. You stop resisting. And right there, they drop one line that quietly rewires the whole subject. When Benedict Gross says matrices represent linear operators, he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: ℝⁿ → ℝⁿ that respects two rules: T(u+v) = T(u) + T(v) T(αu) = αT(u) Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what A represents T means. A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx) = (AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative. T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of linear algebra because it tells you what the course is really about: functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. After that, the rules stop feeling random. Multiplying matrices means doing one move and then another. An inverse means you can undo the move. Eigenvectors are directions that don’t get turned. Changing basis is just describing the same move in a different language. One idea, and a lot of linear algebra suddenly clicks. #LinearAlgebra #Matrices #LinearMaps #Eigenvectors #ChangeOfBasis #Mathematics

Mathelirium

133,454 views • 4 months ago

The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices. Encoding matrices as graphs is a cheat code, making complex behavior simple to study. Let me show you how!

The single most undervalued fact of linear algebra: matrices are graphs, and graphs are matrices. Encoding matrices as graphs is a cheat code, making complex behavior simple to study. Let me show you how!

Tivadar Danka

158,445 views • 6 months ago

Pseudospectral Mirage of Matrices We usually judge a matrix by its eigenvalues, but sometimes the eigenvalues are only the bait. The real monster is hiding in the pseudospectrum. Most of the time, a matrix is shown as a few points in the complex plane called eigenvalues. But for some matrices, especially non-normal matrices, those few points do not tell the whole story. In this scene, the complex plane becomes a moving landscape. The bright anchors are the true eigenvalues. Around them, huge pseudospectral continents grow, split, and collapse as the matrix changes. The moving threads show where the matrix is most sensitive. In those regions, even a tiny change can create a huge response. This is the hidden instability of a matrix made visible. Algebra becomes a pressure map and the pseudospectrum becomes terrain. #Mathematics #LinearAlgebra #SpectralTheory #NumericalAnalysis #MatrixTheory #MathArt #MathematicalArt #STEM #Art

Pseudospectral Mirage of Matrices We usually judge a matrix by its eigenvalues, but sometimes the eigenvalues are only the bait. The real monster is hiding in the pseudospectrum. Most of the time, a matrix is shown as a few points in the complex plane called eigenvalues. But for some matrices, especially non-normal matrices, those few points do not tell the whole story. In this scene, the complex plane becomes a moving landscape. The bright anchors are the true eigenvalues. Around them, huge pseudospectral continents grow, split, and collapse as the matrix changes. The moving threads show where the matrix is most sensitive. In those regions, even a tiny change can create a huge response. This is the hidden instability of a matrix made visible. Algebra becomes a pressure map and the pseudospectrum becomes terrain. #Mathematics #LinearAlgebra #SpectralTheory #NumericalAnalysis #MatrixTheory #MathArt #MathematicalArt #STEM #Art

Mathelirium

28,974 views • 13 days ago

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

The Trap in Every Mathematics Lecture If you’ve taken a lot of math courses, you start to recognize a pattern. There’s a moment where the lecturer is warming up with the obvious stuff...add matrices entrywise, scale by α, do the row-column product...and you’re thinking, alright… where is this going? Then you relax. You stop resisting. And right there, they slip in one line that changes how you see the whole subject. When Benedict Gross says "matrices represent linear operators,"he’s telling you to stop treating a matrix as a rectangle of numbers and start treating it as an action. A linear operator is a function T: Rⁿ → Rⁿ that respects two rules: T(u+v)=T(u)+T(v) and T(αu)=αT(u). Once you pick a basis, T is completely determined by where it sends the basis vectors e₁,…,eₙ. Put T(e₁),…,T(eₙ) into columns and you get a matrix A. That is what "A represents T" means...A is the coordinate portrait of the transformation. Now the punchline that makes matrix multiplication feel inevitable. If B represents S and A represents T, then doing S first and then T is the composition T∘S. In coordinates that becomes A(Bx)=(AB)x. So multiplying matrices is really composing transformations. That’s why multiplication is usually not commutative: T∘S is generally not the same transformation as S∘T, and the matrices inherit that noncommutativity. This explains half of Linear Algebra because it tells you what the course is really about...functions that move vectors around, not grids of numbers. A matrix is just the written form of that function once you choose coordinates. Then the rules stop feeling random Multiplying matrices means doing one move and then another, an inverse means you can undo the move, eigenvectors are directions that don’t get turned, and changing basis is just describing the same move in a different language. That one idea makes a lot of linear algebra click. #LinearAlgebra #Matrices #GroupTheory #GLn #MathLectures #Mathematics

Mathelirium

66,204 views • 5 months ago

This is a great lecture at MIT by David Shirokoff on Markov Chains. He covers the fundamentals of Markov Chains using a simple particle movement example. He starts by explaining how a particle moves between two positions, A & B, with different probabilities. From there, the talk converts the problem into matrix form using a Markov matrix. The main topics covered are: - Transition probabilities - Markov matrices - Probability vectors - Matrix multiplication in Markov Chains - Finding probabilities after n steps - Eigenvalues and eigenvectors - Matrix diagonalization - Long-term steady state distribution

This is a great lecture at MIT by David Shirokoff on Markov Chains. He covers the fundamentals of Markov Chains using a simple particle movement example. He starts by explaining how a particle moves between two positions, A & B, with different probabilities. From there, the talk converts the problem into matrix form using a Markov matrix. The main topics covered are: - Transition probabilities - Markov matrices - Probability vectors - Matrix multiplication in Markov Chains - Finding probabilities after n steps - Eigenvalues and eigenvectors - Matrix diagonalization - Long-term steady state distribution

𝗿𝗮𝗺𝗮𝗸𝗿𝘂𝘀𝗵𝗻𝗮— 𝗲/𝗮𝗰𝗰

140,876 views • 1 month ago

It is possible to recreate how lighting works just by running the dot product between the normal and a vector that represents where the light is coming from #b3d #geometrynodes

It is possible to recreate how lighting works just by running the dot product between the normal and a vector that represents where the light is coming from #b3d #geometrynodes

Aumission

44,151 views • 4 months ago

Introducing Initiatives. Coordinate strategic product efforts and monitor their progress at scale.

Introducing Initiatives. Coordinate strategic product efforts and monitor their progress at scale.

Linear

1,533,154 views • 2 years ago

Introducing Operation Matrix Because Matrix operations generally break kids’ brains on whiteboards… Matrices aren't just numbers- they are transformations in space Now can actually SEE the geometry: every addition, multiplication, inverse, eigenvalue, and eigenvector becomes a transformation in real-time space. What it visualizes: • Basic ops (add/subtract/multiply) • Transpose, Invert, Adjugated • Cross products + Determinant • Matrix powers + Eigenvalues/Eigenvectors #LinearAlgebra #CreativeCoding #ThreeJS #ReactThreeFiber #MathVisualization #EdTech #WebGL #STEM React-three-fiber Three.js

Introducing Operation Matrix Because Matrix operations generally break kids’ brains on whiteboards… Matrices aren't just numbers- they are transformations in space Now can actually SEE the geometry: every addition, multiplication, inverse, eigenvalue, and eigenvector becomes a transformation in real-time space. What it visualizes: • Basic ops (add/subtract/multiply) • Transpose, Invert, Adjugated • Cross products + Determinant • Matrix powers + Eigenvalues/Eigenvectors #LinearAlgebra #CreativeCoding #ThreeJS #ReactThreeFiber #MathVisualization #EdTech #WebGL #STEM React-three-fiber Three.js

SHAHNAB AHMED

58,682 views • 3 months ago

and Iran is the first product

and Iran is the first product

Abier

33,921 views • 4 months ago

Functions are vectors! This perspective lets us apply the tools of linear algebra to computational problems from image and geometry processing to machine learning and light transport—and provides a natural explanation for Fourier series. Let's explore:

Functions are vectors! This perspective lets us apply the tools of linear algebra to computational problems from image and geometry processing to machine learning and light transport—and provides a natural explanation for Fourier series. Let's explore:

Max Slater

30,424 views • 2 years ago

[CLIP] by Hand ✍️ The CLIP (Contrastive Language–Image Pre-training) model, a groundbreaking work by OpenAI, redefines the intersection of computer vision and natural language processing. It is the basis of all the multi-modal foundation models we see today. How does CLIP work? Goal: 🟨 Learn a shared embedding space for text and image [1] Given ↳ A mini batch of 3 text-image pairs ↳ OpenAI used 400 million text-image pairs to train its original CLIP model. Process 1st pair: "big table" [2] 🟪 Text → 2 Vectors (3D) ↳ Look up word embedding vectors using word2vec. [3] 🟩 Image → 2 Vectors (4D) ↳ Divide the image into two patches. ↳ Flatten each patch [4] Process other pairs ↳ Repeat [2]-[3] [5] 🟪 Text Encoder & 🟩 Image Encoder ↳ Encode input vectors into feature vectors ↳ Here, both encoders are simple one layer perceptron (linear + ReLU) ↳ In practice, the encoders are usually transformer models. [6] 🟪 🟩 Mean Pooling: 2 → 1 vector ↳ Average 2 feature vectors into a single vector by averaging across the columns ↳ The goal is to have one vector to represent each image or text [7] 🟪 🟩 -> 🟨 Projection ↳ Note that the text and image feature vectors from the encoders have different dimensions (3D vs. 4D). ↳ Use a linear layer to project image and text vectors to a 2D shared embedding space. 🏋️ Contrastive Pre-training 🏋️ [8] Prepare for MatMul ↳ Copy text vectors (T1,T2,T3) ↳ Copy the transpose of image vectors (I1,I2,I3) ↳ They are all in the 2D shared embedding space. [9] 🟦 MatMul ↳ Multiply T and I matrices. ↳ This is equivalent to taking dot product between every pair of image and text vectors. ↳ The purpose is to use dot product to estimate the similarity between a pair of image-text. [10] 🟦 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [11] 🟦 Softmax: ∑ ↳ Sum each row for 🟩 image→🟪 text ↳ Sum each column for 🟪 text→ 🟩 image [12] 🟦 Softmax: 1 / sum ↳ Divide each element by the column sum to obtain a similarity matrix for 🟪 text→🟩 image ↳ Divide each element by the row sum to obtain a similarity matrix for 🟩 image→🟪 text [13] 🟥 Loss Gradients ↳ The "Targets" for the similarity matrices are Identity Matrices. ↳ Why? If I and T come from the same pair (i=j), we want the highest value, which is 1, and 0 otherwise. ↳ Apply the simple equation of [Similarity - Target] to compute gradients of for both directions. ↳ Why so simple? Because when Softmax and Cross-Entropy Loss are used together, the math magically works out that way. ↳ These gradients kick off the backpropagation process to update weights and biases of the encoders and projection layers (red borders).

[CLIP] by Hand ✍️ The CLIP (Contrastive Language–Image Pre-training) model, a groundbreaking work by OpenAI, redefines the intersection of computer vision and natural language processing. It is the basis of all the multi-modal foundation models we see today. How does CLIP work? Goal: 🟨 Learn a shared embedding space for text and image [1] Given ↳ A mini batch of 3 text-image pairs ↳ OpenAI used 400 million text-image pairs to train its original CLIP model. Process 1st pair: "big table" [2] 🟪 Text → 2 Vectors (3D) ↳ Look up word embedding vectors using word2vec. [3] 🟩 Image → 2 Vectors (4D) ↳ Divide the image into two patches. ↳ Flatten each patch [4] Process other pairs ↳ Repeat [2]-[3] [5] 🟪 Text Encoder & 🟩 Image Encoder ↳ Encode input vectors into feature vectors ↳ Here, both encoders are simple one layer perceptron (linear + ReLU) ↳ In practice, the encoders are usually transformer models. [6] 🟪 🟩 Mean Pooling: 2 → 1 vector ↳ Average 2 feature vectors into a single vector by averaging across the columns ↳ The goal is to have one vector to represent each image or text [7] 🟪 🟩 -> 🟨 Projection ↳ Note that the text and image feature vectors from the encoders have different dimensions (3D vs. 4D). ↳ Use a linear layer to project image and text vectors to a 2D shared embedding space. 🏋️ Contrastive Pre-training 🏋️ [8] Prepare for MatMul ↳ Copy text vectors (T1,T2,T3) ↳ Copy the transpose of image vectors (I1,I2,I3) ↳ They are all in the 2D shared embedding space. [9] 🟦 MatMul ↳ Multiply T and I matrices. ↳ This is equivalent to taking dot product between every pair of image and text vectors. ↳ The purpose is to use dot product to estimate the similarity between a pair of image-text. [10] 🟦 Softmax: e^x ↳ Raise e to the power of the number in each cell ↳ To simplify hand calculation, we approximate e^□ with 3^□. [11] 🟦 Softmax: ∑ ↳ Sum each row for 🟩 image→🟪 text ↳ Sum each column for 🟪 text→ 🟩 image [12] 🟦 Softmax: 1 / sum ↳ Divide each element by the column sum to obtain a similarity matrix for 🟪 text→🟩 image ↳ Divide each element by the row sum to obtain a similarity matrix for 🟩 image→🟪 text [13] 🟥 Loss Gradients ↳ The "Targets" for the similarity matrices are Identity Matrices. ↳ Why? If I and T come from the same pair (i=j), we want the highest value, which is 1, and 0 otherwise. ↳ Apply the simple equation of [Similarity - Target] to compute gradients of for both directions. ↳ Why so simple? Because when Softmax and Cross-Entropy Loss are used together, the math magically works out that way. ↳ These gradients kick off the backpropagation process to update weights and biases of the encoders and projection layers (red borders).

Tom Yeh

67,790 views • 2 years ago

Introducing Product Intelligence. AI-assisted product operations for routine, manual tasks. • Automates the overhead of triage intake • Detects duplicates and related work • Suggests the right assignee, team, labels, and more Now available in Technology Preview.

Introducing Product Intelligence. AI-assisted product operations for routine, manual tasks. • Automates the overhead of triage intake • Detects duplicates and related work • Suggests the right assignee, team, labels, and more Now available in Technology Preview.

Linear

190,481 views • 11 months ago

Linear: puts the actual product on the homepage and lets you click through it. More companies should be doing this.

Linear: puts the actual product on the homepage and lets you click through it. More companies should be doing this.

Brett

132,437 views • 4 months ago

Introducing Alloy interactive capture. > Open your product in the browser and start capturing > Click through pages or flows to create an interactive prototype > Prompt your product idea and instantly share a link Available today for all Alloy users.

Introducing Alloy interactive capture. > Open your product in the browser and start capturing > Click through pages or flows to create an interactive prototype > Prompt your product idea and instantly share a link Available today for all Alloy users.

Simon Kubica

12,046 views • 5 months ago

Introducing Refix - the world's first revenue obsessed AI Product Manager It continuously watches product, marketing, sales, support metrics; finds revenue opportunities proactively, and ships autonomously For the first time, your product can improve itself.

Introducing Refix - the world's first revenue obsessed AI Product Manager It continuously watches product, marketing, sales, support metrics; finds revenue opportunities proactively, and ships autonomously For the first time, your product can improve itself.

Neil Agarwal

49,834 views • 2 months ago

Introducing Index – the new planning tool for Product Managers on Linear. ➡️ Product leaders have long deserved a modern experience for product planning and discovery. Something powerful, flexible, and integrated with the latest tools. That’s why we built Index (Index), the new standard for Product Management: 1. Every layout at your fingertips Breeze through planning with a whiteboard, table, roadmap, or board available every step of the way. One place to go, infinite flexibility, and context on your projects carried throughout. 2. Built for B2B Product Management Natively connect to your customer context, leveraging their requests, insights, and feedback all in one place. Index connects to Hubspot, Intercom, Dovetail, Attio, Gong, and more – so you can back up your vision with customer evidence. 3. The product-native companion to Linear To build a true companion to Linear for PMs, we knew we had to raise the bar. Interactions are instant, every layout is collaborative, and it fits Linear like a glove with native sync. We're live today, and you can pre-order now at

Introducing Index – the new planning tool for Product Managers on Linear. ➡️ Product leaders have long deserved a modern experience for product planning and discovery. Something powerful, flexible, and integrated with the latest tools. That’s why we built Index (Index), the new standard for Product Management: 1. Every layout at your fingertips Breeze through planning with a whiteboard, table, roadmap, or board available every step of the way. One place to go, infinite flexibility, and context on your projects carried throughout. 2. Built for B2B Product Management Natively connect to your customer context, leveraging their requests, insights, and feedback all in one place. Index connects to Hubspot, Intercom, Dovetail, Attio, Gong, and more – so you can back up your vision with customer evidence. 3. The product-native companion to Linear To build a true companion to Linear for PMs, we knew we had to raise the bar. Interactions are instant, every layout is collaborative, and it fits Linear like a glove with native sync. We're live today, and you can pre-order now at

Simon Kubica

66,073 views • 1 year ago

introducing matrix. the ultimate multi-agent runtime that turns your product into actual revenue and automates 99% of what it takes to run a company. probably the most different, most future-feeling agent product you've ever seen. the harness, the interface, everything is rebuilt from scratch. think of it as the complement to codex and claude code. they write the code. matrix does literally everything else you actually need done. plug in your codex subscription and run it for free, or byok. fully free either way. this video was edited by matrix and voiced by an agent. more real cases coming soon. matrix is in beta preview right now. full launch next week. retweet + comment and we'll send invite codes to the first 200.

introducing matrix. the ultimate multi-agent runtime that turns your product into actual revenue and automates 99% of what it takes to run a company. probably the most different, most future-feeling agent product you've ever seen. the harness, the interface, everything is rebuilt from scratch. think of it as the complement to codex and claude code. they write the code. matrix does literally everything else you actually need done. plug in your codex subscription and run it for free, or byok. fully free either way. this video was edited by matrix and voiced by an agent. more real cases coming soon. matrix is in beta preview right now. full launch next week. retweet + comment and we'll send invite codes to the first 200.

Derek Nee

15,094 views • 18 days ago

It's Friday, and that means... a new Billing product update!! "Billing" will be the first product from the Clerk Commerce team! Coming soon!

It's Friday, and that means... a new Billing product update!! "Billing" will be the first product from the Clerk Commerce team! Coming soon!

Drew Wilson

18,375 views • 1 year ago