systems. He introduces the (intersecting lines) and the Column Picture (combining vectors). Understanding the Column Picture is the "aha!" moment for most students. 2. Matrix Multiplication and Factorization
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Ax = b (no solution) ↓ Minimize ||Ax - b||^2 ↓ Derivative = 0 → A^T A x̂ = A^T b ↓ If columns independent → x̂ = (A^T A)^-1 A^T b ↓ Projection p = A x̂ systems
The are the first non-zero entries in each row after elimination. For an (n \times n) matrix: For an (n \times n) matrix: How do
How do you solve a system of equations that has no solution? This is the heart of data science and statistics. Strang’s notes on and the Gram-Schmidt process provide the tools to find the "best possible" answer. 5. Determinants and Eigenvalues
Computation method: