Values with direction
A vector is an ordered list of numbers. It can represent a point, a displacement, a signal, or any collection of related measurements.
v = (v1, v2, …, vn)Library 01 · Mathematics
Linear algebra
The mathematics of vectors, spaces, and transformations—an economical language for describing direction, structure, and change.
Vectors encode magnitude and direction
Three foundational objects
Linear algebra starts with quantities that can be added and scaled. From that modest rule comes a framework used in geometry, computing, statistics, physics, economics, and machine learning.
Numbers in a grid
A matrix organises numbers into rows and columns. It can store data or encode a rule that transforms one vector into another.
A = [aij]Structure-preserving maps
A linear transformation respects vector addition and scalar multiplication. Matrices give these transformations a concrete form.
T(v) = AvThe unifying idea
Preserve the linear combinations.
A transformation is linear when combining inputs before the transformation gives the same result as transforming each input and then combining the outputs. This single property makes large systems predictable and computable.
T(αv + βw) = αT(v) + βT(w)
A map of the subject
Ideas in linear algebra build on one another. Begin with matrices and systems, then move toward vector spaces, eigenvalues, and decompositions.
01
Identity matrixThe matrix equivalent of multiplying by one.
02
Systems of equationsRepresent and solve simultaneous constraints.
03
Vector spacesDescribe the settings in which vectors live.
04
EigenvaluesFind directions a transformation does not turn.