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

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ℏεsam

72,907 subscribers

63,641 Aufrufe • vor 1 Jahr •via X (Twitter)

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Profilbild von ℏεsam
ℏεsamvor 1 Jahr

🔗

Profilbild von Sayan Chakraborty
Sayan Chakrabortyvor 1 Jahr

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

Profilbild von steve ike
steve ikevor 1 Jahr

Love the idea of interactive textbooks.

Profilbild von Aditya Dixit
Aditya Dixitvor 1 Jahr

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

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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
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