An Illustrated Introduction to Linear Algebra (www.ducktyped.org)

This chapter introduces two foundational ways to think about linear systems — Gaussian elimination and the “row” versus “column” pictures — using simple, tangible examples (nickels & pennies, milk & bread with carbs/protein). Gaussian elimination is presented as the algebraic recipe for removing variables (subtract multiples of equations) until you isolate one unknown; the row picture translates each equation into a geometric hyperplane (lines in 2D) and finds the unique solution at their intersection. The post emphasizes that elimination is an old, practical technique separate from the broader framework of linear algebra. The column picture reframes the same problem: coefficients are vectors and the target is a vector, so solving the system becomes expressing the target as a linear combination of column vectors. Graphically you add scaled arrows (scalar·vector = elementwise scaling; vector addition is elementwise) until you reach the target point. The chapter closes by showing the matrix×vector notation and teases the dot product next. For the AI/ML community this dual intuition matters: vectors and matrices are the language of features, weights, embeddings and linear layers; understanding row (constraints/hyperplanes) and column (span/linear combinations) views clarifies why matrix multiplication, dot products, least-squares solutions and numerical methods underlie model training, dimensionality, and representation learning.