Linear Transformation

6 min read#linear-algebra

A way of warping space that keeps grid lines straight, evenly spaced, and pinned to a fixed origin — exactly what a matrix encodes.

Contents

Linear Transformation

A linear transformation is a function T that takes vectors to vectors while preserving the two operations that define a vector space:

T(\vec u + \vec v) = T(\vec u) + T(\vec v), \qquad T(c\,\vec v) = c\,T(\vec v).

Geometrically these algebraic rules have a vivid consequence: grid lines stay straight and evenly spaced, and the origin never moves. Space can rotate, stretch, shear, or flip — but it cannot bend or tear. Every such transformation of the plane is captured by a single 2\times2 Matrix, whose two columns are simply where the transformation sends the basis vectors (1,0) and (0,1).

Drag the basis vectors

Below is the identity grid warped by a live matrix. The two arrows are the matrix's columns — the landing spots of the x- and y-axes. Grab either arrowhead and drag: the entire grid, and the shaded unit square with its F, deform to follow. Notice what cannot happen — lines never curve, parallel lines stay parallel, and the origin stays put. That is linearity, made visible.

Drag the blue or amber arrowhead to set the matrix columns. The grid, unit square, and F warp accordingly; the origin stays fixed. Watch the determinant flip sign when you turn the square inside-out.

Why columns are everything

Because T respects linear combinations, knowing what it does to the basis vectors fixes what it does to all vectors. If \vec v = x\,\hat\imath + y\,\hat\jmath, then T(\vec v) = x\,T(\hat\imath) + y\,T(\hat\jmath) — and T(\hat\imath), T(\hat\jmath) are exactly the columns of the matrix. This is why "apply a matrix" and "apply a linear transformation" mean the same thing, and why composing transformations is just Matrix Multiplication.

Reading the geometry

See also