gdpark.blog
한국어

Series

Linear Algebra I Studied

23 posts

  1. #1

    Prologue: Heralding the Beginning

    Stumbled onto a hidden-gem linear algebra course on educast.pro while cramming for quantum mechanics, and the pure love-of-learning vibe had me completely hooked.

    · 2 min read
  2. #1

    Vector Spaces

    Turns out "linear" literally just means line-shaped — lol — and here's why that embarrassingly simple idea is the backbone of basically all of math and STEM.

    · 5 min read
  3. #2

    Linear Dependence and Linear Independence

    Breaking down linear combinations, span, and what it actually means for vectors to be linearly independent — all in plain terms.

    · 4 min read
  4. #3

    Basis

    So a basis is literally the intersection of spanning sets and linearly independent sets — and there are TONS of them, plus order actually matters!!

    · 4 min read
  5. #4

    Matrices and Matrix Multiplication

    We dive into linear algebra by unpacking functions as set mappings, then zoom in on linear mappings and how they all secretly boil down to matrices.

    · 6 min read
  6. #5

    Surjective, Injective, and Bijective Functions

    We nail down surjective, injective, and bijective functions — and prove why same-dimension linear maps only need one condition to guarantee the other.

    · 7 min read
  7. #6

    Kernel and Image

    We nail down the kernel and image of a linear map, see what injective and surjective really mean, and lock in the theorem that links them all!!!!

    · 2 min read
  8. #7

    Isomorphism and the Dimension Theorem

    Kernel and image lead us to isomorphism — two vector spaces with the same dimension and a bijective map between them are literally just the same space in disguise!!!!

    · 7 min read
  9. #8

    Change of Basis

    Same linear map, totally different matrix depending on your basis — let's dig into why that happens and how A and A' are actually related.

    · 6 min read
  10. #9

    Systems of Linear Equations and Gaussian Elimination

    A casual walkthrough of systems of linear equations — from middle-school basics all the way up to matrix form, homogeneous vs. non-homogeneous, and Gaussian elimination.

    · 10 min read
  11. #10

    Elementary Row Operation Matrices (EROM)

    Back at it with the general solution of AX = B — recapping RREF, breaking down dependent vs. independent variables, and chasing down X_0 and ker A.

    · 9 min read
  12. #11

    The Rank Theorem

    We poke at whether RREF is unique, then laser in on invertible matrices to show their RREF has to be the identity — and that's the Rank Theorem taking shape!

    · 9 min read
  13. #12

    Multilinear Forms, Alternating Forms, and the Epsilon Symbol

    We finally tackle *how* to tell if a matrix is invertible by wading through multilinear forms, alternating forms, and the epsilon symbol — all the groundwork for determinants!

    · 9 min read
  14. #13

    Similar Matrices

    Before we can actually crunch a determinant, we load up on similar matrices — plus why det A = 0 is the snap-decision test for invertibility.

    · 6 min read
  15. #14

    Determinants

    We finally crack the determinant open — from 2x2 all the way to 4x4 — by tracing it straight back to where it really came from: the Alternating Form, lol.

    · 7 min read
  16. #15

    The Adjoint Matrix

    We dig into triangular and Vandermonde matrices, crack their determinants, then build up to the adjoint matrix — which is about to become super important for what's coming next.

    · 6 min read
  17. #16

    Block Matrices

    A casual dive into diagonalization — what it means to turn a matrix diagonal, why that's such a big deal in physics, and how blocking gets the whole thing started.

    · 9 min read
  18. #17

    Introduction to Diagonalization, Eigenvalues, and Eigenvectors

    Finally cracking how to diagonalize a matrix without hunting for the right basis — turns out eigenvalues and eigenvectors are the secret the whole time!

    · 6 min read
  19. #18

    Characteristic Polynomials and the Cayley–Hamilton Theorem

    We chase down which matrices can actually be diagonalized — starting with characteristic polynomials, hitting complex eigenvalues, and landing on the Cayley–Hamilton theorem.

    · 12 min read
  20. #19

    Number Theory and Polynomial Background for the Decomposition Theorem

    A casual walk through naturals, integers, divisibility, and the division algorithm, building up to why any subset of Z closed under +, −, and × must look like nZ.

    · 13 min read
  21. #20

    The First Decomposition Theorem

    We revisit diagonalization up close — eigenvalues, kernels, and how a 2D space splits into two 1D invariant subspaces — as a warm-up for the full Decomposition Theorem.

    · 8 min read
  22. #21

    Minimal Polynomials, Companion Matrices, and More

    Diving into minimal polynomials and their sneaky relationship with characteristic polynomials — turns out it's the same divisor-hunting logic we used back in number theory!!!!

    · 16 min read
  23. #22

    Is Further Block Decomposition Always Possible?

    Poking at a special case where the minimal and characteristic polynomials match as a power of an irreducible polynomial, and what that tells us about block decomposition.

    · 5 min read