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As a newly minted Ph.D., Paul Halmos came to the Institute for Advanced Study in 1938--even though he did not have a fellowship--to study among the many giants of mathematics who had recently joined the faculty. He eventually became John von Neumann's research assistant, and it was one of von Neumann's inspiring lectures that spurred Halmos to write Finite Dimensional Vector Spaces. The book brought him instant fame as an expositor of mathematics. Finite Dimensional Vector Spaces combines algebra and geometry to discuss the three-dimensional area where vectors can be plotted. The book broke ground as the first formal introduction to linear algebra, a branch of modern mathematics that studies vectors and vector spaces. The book continues to exert its influence sixty years after publication, as linear algebra is now widely used, not only in mathematics but also in the natural and social sciences, for studying such subjects as weather problems, traffic flow, electronic circuits, and population genetics. In 1983 Halmos received the coveted Steele Prize for exposition from the American Mathematical Society for "his many graduate texts in mathematics dealing with finite dimensional vector spaces, measure theory, ergodic theory, and Hilbert space."

Transformations (Mathematics) --- Generalized spaces. --- Absolute value. --- Accuracy and precision. --- Addition. --- Affine space. --- Algebraic closure. --- Algebraic equation. --- Algebraic operation. --- Algebraically closed field. --- Associative property. --- Automorphism. --- Axiom. --- Banach space. --- Basis (linear algebra). --- Bilinear form. --- Bounded operator. --- Cardinal number. --- Cayley transform. --- Characteristic equation. --- Characterization (mathematics). --- Coefficient. --- Commutative property. --- Complex number. --- Complex plane. --- Computation. --- Congruence relation. --- Convex set. --- Coordinate system. --- Determinant. --- Diagonal matrix. --- Dimension (vector space). --- Dimension. --- Dimensional analysis. --- Direct product. --- Direct proof. --- Direct sum. --- Division by zero. --- Dot product. --- Dual basis. --- Eigenvalues and eigenvectors. --- Elementary proof. --- Equation. --- Euclidean space. --- Existential quantification. --- Function of a real variable. --- Functional calculus. --- Fundamental theorem. --- Geometry. --- Gram–Schmidt process. --- Hermitian matrix. --- Hilbert space. --- Infimum and supremum. --- Jordan normal form. --- Lebesgue integration. --- Linear combination. --- Linear function. --- Linear independence. --- Linear map. --- Linear programming. --- Linearity. --- Manifold. --- Mathematical induction. --- Mathematics. --- Minimal polynomial (field theory). --- Minor (linear algebra). --- Monomial. --- Multiplication sign. --- Natural number. --- Nilpotent. --- Normal matrix. --- Normal operator. --- Number theory. --- Orthogonal basis. --- Orthogonal complement. --- Orthogonal coordinates. --- Orthogonality. --- Orthonormality. --- Polynomial. --- Quotient space (linear algebra). --- Quotient space (topology). --- Real number. --- Real variable. --- Scalar (physics). --- Scientific notation. --- Series (mathematics). --- Set (mathematics). --- Sign (mathematics). --- Special case. --- Spectral theorem. --- Spectral theory. --- Summation. --- Tensor calculus. --- Theorem. --- Topology. --- Transitive relation. --- Unbounded operator. --- Uncountable set. --- Unit sphere. --- Unitary transformation. --- Variable (mathematics). --- Vector space.