Short answers, pulled from the story.
Giuseppe Peano published the definition of vector spaces in 1888. He established the modern framework by requiring these structures to satisfy eight specific rules governing addition and scalar multiplication.
René Descartes and Pierre de Fermat identified solutions to equations with points on curves around 1636 when the concept emerged from analytic geometry. Bernard Bolzano introduced operations on points, lines, and planes in 1804 that served as predecessors to modern vectors.
A basis is a subset of a vector space whose elements are linearly independent and span the entire space. All bases of a given vector space share the same cardinality known as the dimension of that space.
Eigenvalues and eigenvectors arise when comparing vectors with their images under endomorphisms where any nonzero vector satisfying Tv equals lambda times v qualifies as an eigenvector with eigenvalue lambda. The characteristic polynomial determines these values through its roots.
Stefan Banach constructed function spaces later formalized by Stefan Banach and David Hilbert around 1920. This period marked the interaction between algebra and functional analysis through key concepts such as p-integrable functions and Hilbert spaces.