Short answers, pulled from the story.
James Joseph Sylvester coined the word matrix in 1848 from the Latin term for womb. This naming established a metaphor that defined the structure of modern mathematics. Before this date, the manipulation of numbers in rectangular arrays was a chaotic collection of techniques used by astronomers and surveyors.
René Descartes published his method for linking algebra and geometry in 1637. This publication marked a violent rupture in the history of geometry by introducing the idea that points in space could be represented by sequences of numbers. The resulting Cartesian geometry allowed mathematicians to describe complex spatial relationships using simple arithmetic operations.
Giuseppe Peano introduced the precise definitions of vector spaces in 1888. This development occurred during a period when the telegraph and the need to model complex physical phenomena forced mathematicians to refine their tools. The transition from abstract theory to practical necessity followed the publication of James Clerk Maxwell's 1873 Treatise on Electricity and Magnetism.
Arthur Cayley introduced matrix multiplication in 1856. This pivotal moment transformed matrices from static arrays into dynamic operators capable of being composed and manipulated as single objects. Cayley's decision to use a single letter to denote a matrix treated the entire array as an aggregate object rather than a collection of individual numbers.
The discovery of eigenvalues and eigenvectors was made in the 19th century. An eigenvector is a nonzero vector that remains in the same direction after a linear transformation, scaled only by a scalar factor known as an eigenvalue. This concept allowed mathematicians to simplify complex transformations by finding a basis of eigenvectors through a process known as diagonalization.
The 20th century witnessed the explosive growth of linear algebra as it became the primary language of quantum mechanics and functional analysis. The study of function spaces such as Hilbert spaces and Banach spaces relied heavily on the principles of linear algebra to describe the behavior of wave functions. The Gram, Schmidt procedure provided a method for simplifying complex calculations in quantum theory during this period.