Index

Definitions and theorems of Linear algebra and analytic geometry

The 93 definitions and 109 theorems of the theory of Linear algebra and analytic geometry, in the order of the topics. Each entry links to its box in the theory page.

Vectors in the plane and in space

Definition Geometric vector
Definition Sum of vectors
Definition Multiplication by a scalar
Definition Components of a vector
Theorem Operations in components
Theorem Vector between two points and midpoint
Definition Linear combination
Definition Magnitude of a vector
Definition Distance between two points
Definition Unit vector
Definition Angle between two vectors
Definition Dot product
Theorem Dot product in components
Theorem Properties of the dot product
Definition Orthogonal vectors
Theorem Orthogonal projection onto a vector
Definition Cross product
Theorem Algebraic properties of the cross product
Theorem Geometric meaning of the cross product
Theorem Area of the parallelogram and of the triangle
Definition Parallel vectors
Theorem Parallelism criterion
Theorem Collinear points
Definition Scalar triple product
Theorem Volume of the parallelepiped and of the tetrahedron
Definition Coplanar vectors
Theorem Coplanarity criterion

Matrices and matrix operations

Definition Matrix
Definition Square matrices, row vectors and column vectors
Definition Diagonal, triangular and identity matrices
Definition Symmetric and skew-symmetric matrices
Definition Sum and multiplication by a scalar
Theorem Properties of the sum and of multiplication by a scalar
Definition Row-by-column product
Theorem Properties of the row-by-column product
Definition Commuting matrices
Definition Zero divisors
Definition Transpose matrix
Theorem Properties of the transpose
Definition Powers of a square matrix
Theorem Powers of diagonal and triangular matrices
Definition Nilpotent matrix
Definition Invertible matrix
Theorem Uniqueness of the inverse
Theorem Properties of the inverse matrix
Theorem Sufficient conditions for non-invertibility
Definition Elementary row operations
Definition Row echelon matrix and pivots
Theorem Gaussian elimination
Theorem Elementary matrices
Theorem Gauss-Jordan theorem
Theorem Right inverse and left inverse

Determinant and inverse matrix

Definition Determinant of order 2
Definition Determinant of order 3 and Sarrus' rule
Definition Minor and cofactor
Definition Determinant of order n
Theorem Laplace's theorem
Theorem Properties of the determinant
Theorem Determinant of a triangular matrix
Theorem Effect of elementary operations
Theorem Determinant of the product by an elementary matrix
Theorem Invertibility criterion
Theorem Binet's theorem
Theorem Determinant of the inverse
Definition Cofactor matrix
Theorem Formula for the inverse

Linear systems and rank

Definition Linear system
Definition Consistent and inconsistent system
Definition Matrix form
Definition Elementary row operations
Theorem Elementary operations and equivalent systems
Definition Minor of a matrix
Definition Rank
Theorem Rank and reduction to echelon form
Definition Bordered minor
Theorem Bordered minors theorem (Kronecker)
Theorem Rouché-Capelli theorem
Theorem Cramer's rule
Definition Homogeneous system
Theorem Nontrivial solutions of a homogeneous system
Theorem Structure of the solutions

Vector spaces, bases and dimension

Definition Real vector space
Definition Vector subspace
Theorem Subspace criterion
Definition Linear combination
Theorem Subspace spanned
Definition Spanning set
Theorem Removal of a redundant generator
Definition Linearly independent vectors
Theorem Dependence and linear combinations
Theorem Independence and rank
Theorem Adding a vector to an independent set
Definition Basis
Theorem Uniqueness of coordinates
Definition Coordinates with respect to a basis
Theorem Steinitz lemma
Theorem Invariance of the number of elements of a basis
Definition Dimension
Theorem Consequences of the dimension
Theorem Dimension of a subspace
Theorem Extracting a basis
Theorem Completion to a basis
Definition Parametric form
Definition Cartesian form
Theorem Intersection of subspaces
Definition Sum of subspaces
Theorem Grassmann's formula
Definition Direct sum
Theorem Uniqueness of the decomposition

Linear maps

Definition Linear map
Theorem First properties
Theorem Existence and uniqueness on the vectors of a basis
Definition Associated matrix
Theorem Coordinates of the image
Definition Kernel and image
Theorem Kernel and image are subspaces
Theorem Generators of the image and rank
Theorem Injectivity and kernel
Theorem Surjectivity and rank
Theorem Injective maps and independent vectors
Theorem Rank-nullity theorem
Theorem Consequences of the rank-nullity theorem
Definition Isomorphism
Theorem Isomorphic spaces and dimension
Theorem Isomorphisms and determinant
Theorem Matrix of the composition
Theorem Matrix of the inverse map
Definition Change-of-basis matrix
Theorem Change-of-basis formula
Definition Similar matrices
Theorem Similarity invariants

Eigenvalues, eigenvectors and diagonalization

Definition Eigenvalue and eigenvector of an endomorphism
Definition Eigenvalue and eigenvector of a matrix
Definition Characteristic polynomial
Theorem Eigenvalues as roots of the characteristic polynomial
Theorem Trace, determinant and eigenvalues
Theorem Similar matrices have the same characteristic polynomial
Definition Eigenspace
Definition Algebraic and geometric multiplicity
Theorem Relation between the multiplicities
Theorem Eigenvectors associated with distinct eigenvalues
Definition Diagonalizable matrix and endomorphism
Theorem Diagonalizability and bases of eigenvectors
Theorem Diagonalizability criterion
Theorem Distinct real eigenvalues
Theorem Powers of a diagonalizable matrix

Inner product, orthogonality and the spectral theorem

Definition Standard scalar product
Theorem Properties of the scalar product
Definition Norm and distance
Theorem Cauchy-Schwarz inequality
Theorem Triangle inequality
Definition Angle between two vectors
Definition Orthogonal vectors
Theorem Pythagorean theorem
Theorem Independence of orthogonal vectors
Definition Orthonormal basis
Theorem Coordinates with respect to an orthonormal basis
Definition Orthogonal projection onto a vector
Theorem Orthogonal projection onto a subspace
Theorem Distance from a subspace
Theorem Gram-Schmidt process
Definition Orthogonal complement
Theorem Dimension of the orthogonal complement
Definition Orthogonal matrix
Theorem Properties of orthogonal matrices
Theorem Eigenvalues and eigenvectors of a symmetric matrix
Theorem Real spectral theorem

Lines and planes in space

Definition Plane through a point orthogonal to a vector
Theorem Cartesian equation of the plane
Definition Parametric equations of the plane
Definition Parametric equations of the line
Definition Cartesian equations of the line
Theorem Relative position of two planes
Theorem Relative position of a line and a plane
Definition Coplanar lines and skew lines
Theorem Coplanarity criterion for two lines
Definition Angle between two lines
Definition Angle between a line and a plane
Definition Angle between two planes
Theorem Distance from a point to a plane
Theorem Distance from a point to a line
Definition Orthogonal projection of a point onto a plane
Definition Symmetric point
Definition Pencil of planes
Theorem Planes containing a line
Theorem Common perpendicular to two skew lines
Theorem Distance between two skew lines

Conics and quadrics

Definition Conic
Definition Matrix associated with a conic
Definition Rigid motion
Theorem Orthogonal invariants
Theorem Canonical form of central conics
Theorem Canonical form of the parabola
Definition Degenerate conic
Theorem Classification of non-degenerate conics
Definition Center of a conic
Theorem Classification of degenerate conics
Definition Quadric and associated matrix
Theorem Canonical form of quadrics
Definition Non-degenerate quadrics
Definition Cones and cylinders
Theorem Recognition of non-degenerate quadrics