math / Linear Algebra / free

Vectors & Matrices

Move a vector through a transformation and track what survives.

Follow the mechanism1 / 3

A vector carries direction and magnitude. Its coordinates describe a directed quantity in a chosen basis.

linear transformationmatrix changes space

Focus the idea

matrix A[ 1.6 0.0 ; 0.0 0.8 ]

input v(2.0, 1.0)

output Av(3.2, 0.8)

area scale det A1.3

Transformation

Vector x 2.0Vector y 1.0

The solid grid is the dashed grid after A acts on it: stretch moves every grid line along with the basis vectors Ae₁ and Ae₂.

A matrix transforms every vector in the same space. The determinant tells you how the transformation changes area and whether it flips orientation; collapse has determinant zero and loses a dimension.