What is Linear Alebgra?

Linear algebra is the study of vectors (quantities with direction and magnitude) and linear transformations (rules that stretch, shrink, rotate or reflect vectors) using matrices (grids of numbers).

• Vectors are like arrows in space, telling you “which way” and “how far.”

• Matrices are machines: you feed in a vector, and they output another vector, often transforming it in simple, predictable ways (scaling, rotating, shearing).

• Linear means these machines obey two rules:

  1. Additivity: transforming the sum of two vectors equals the sum of their transformations.

  2. Homogeneity: scaling a vector before or after transformation gives the same result.

Linear algebra lets you break complex problems into simple pieces—you represent data or operations as vectors and matrices, then use neat, powerful tools (like solving matrix equations or finding “best-fit” lines) to analyze, compute, and understand patterns in anything from graphics to machine learning to physics.