Linear algebra

(the mathematics of vectors, matrices, and linear transformations)

What is linear algebra?

Linear algebra is the study of vectors (quantities with both size and direction) and matrices (organized grids of numbers), and how they combine to describe things like rotations, scaling, and systems of equations, all at once.

Vectors: size and direction together

A vector can be written as a pair of numbers, like (3, 2), representing a step of 3 units right and 2 units up. Adding two vectors adds their matching parts: (3, 2) + (1, 4) = (4, 6) — you can picture this as taking one step, then the other, and ending up at the combined position. Scalar multiplication stretches a vector: 2 × (3, 2) = (6, 4), doubling its length while keeping its direction.

Matrices: grids that transform vectors

A matrix is a rectangular grid of numbers. One of its most useful jobs is transforming vectors — taking an input vector and producing a new output vector. For example, the matrix that swaps a vector's two coordinates takes (3, 2) to (2, 3). Different matrices produce different transformations: stretching, rotating, flipping, or combinations of these.

Solving two equations at once, using a matrix

The system of equations 2x + y = 8 and x − y = 1 can be written using a matrix that captures both equations' coefficients at once, and solving the system means finding the vector (x, y) that the matrix transforms into (8, 1). Substitution (from algebra) gives the same answer here: from the second equation, y = x − 1; substituting into the first gives 2x + (x−1) = 8, so 3x = 9, x = 3, and y = 2. Matrices don't do anything algebra can't already do for a simple 2-equation system — their real advantage shows up once there are many equations and many unknowns at once.

A real proof: matrix order matters

With ordinary numbers, 2 × 3 always equals 3 × 2 — multiplication order never matters. With matrices, it usually does. Here's a concrete demonstration that this isn't just a technicality: take the matrix that rotates a vector 90 degrees, and the matrix that flips a vector's horizontal direction. Apply "rotate, then flip" to the vector (1, 0): rotating gives (0, 1), and then flipping gives (0, 1) still (flipping only changes the horizontal part, which is already 0). Now apply "flip, then rotate" to the same starting vector (1, 0): flipping first gives (−1, 0), and then rotating gives (0, −1). The two results, (0, 1) and (0, −1), are different — so doing the two transformations in a different order genuinely produces a different outcome, proving that matrix "multiplication" (applying one transformation after another) does not generally commute.

Why linear algebra matters

Linear algebra is the mathematical backbone behind computer graphics (every rotation and zoom on a screen is a matrix transformation), machine learning (which represents data as vectors and learning as matrix operations), and any field that deals with many related quantities changing together at once, from economics to engineering.