Learn Year 1
Linear algebra
315 ideas in teaching order. Each idea comes after the ideas that it builds on. 312 of them have a page so far; the others are on the list in grey.
- Linear equation definition
- System of linear equations definition
- Homogeneous system of linear equations definition
- Consistent and inconsistent systems definition
- Coefficient matrix definition
- Augmented matrix definition
- Elementary row operations definition
- Row equivalence definition
- Row echelon form definition
- Reduced row echelon form definition
- Pivot definition
- Gaussian elimination concept
- Gauss–Jordan elimination concept
- Back substitution concept
- Rouché–Capelli theorem statement
- Matrix definition
- Square matrix definition
- Zero matrix object
- Matrix addition definition
- Scalar multiplication of matrices definition
- Matrix–vector product definition
- Matrix multiplication definition
- Associativity of matrix multiplication statement
- Distributivity of matrix multiplication statement
- Non-commutativity of matrix multiplication counterexample
- Identity matrix object
- Matrix ring object
- Diagonal matrix definition
- Scalar matrix definition
- Upper triangular matrix definition
- Lower triangular matrix definition
- Transpose definition
- Transpose of a product statement
- Symmetric matrix definition
- Skew-symmetric matrix definition
- Conjugate transpose definition
- Hermitian matrix definition
- Block matrix definition
- Block diagonal matrix definition
- Trace of a matrix definition
- Trace is linear statement
- Trace of a transpose statement
- Trace of a product is symmetric statement
- Invertible matrix definition
- Inverse of a product statement
- One-sided inverse of a square matrix is two-sided statement
- General linear group object
- Elementary matrix definition
- Transvection definition
- Permutation matrix definition
- Computing the inverse by row reduction concept
- LU decomposition concept
- Nilpotent matrix definition
- Idempotent matrix definition
- Circulant matrix example
- Vector space definition
- Coordinate space object
- Space of matrices example
- Space of polynomials of bounded degree example
- Function space example
- Product of vector spaces definition
- Linear subspace definition
- Intersection of subspaces statement
- Sum of subspaces definition
- Linear combination definition
- Linear span definition
- Spanning set definition
- Linear independence definition
- Basis definition
- Standard basis object
- Every vector space has a basis statement
- Every spanning set contains a basis statement
- Every linearly independent set extends to a basis statement
- Steinitz exchange lemma statement
- Finite-dimensional vector space definition
- Dimension definition
- Dimension theorem for vector spaces statement
- Dimension of the coordinate space statement
- Dimension of a subspace statement
- A subspace of full dimension is the whole space statement
- Linearly independent set of the right size is a basis statement
- Grassmann's formula statement
- Coordinate vector definition
- Direct sum of subspaces definition
- External direct sum definition
- Dimension of a direct product statement
- Complementary subspaces definition
- Existence of a complementary subspace statement
- quotient space definition
- Quotient map definition
- Dimension of a quotient space statement
- Linear map definition
- Differentiation as a linear map example
- Composition of linear maps definition
- Kernel of a linear map definition
- Image of a linear map definition
- A linear map is injective iff its kernel is trivial statement
- A linear map is determined by its values on a basis statement
- Rank of a linear map definition
- Nullity definition
- Rank–nullity theorem statement
- Linear isomorphism definition
- Finite-dimensional spaces are isomorphic iff they have equal dimension statement
- Isomorphism with the coordinate space statement
- An injective endomorphism of a finite-dimensional space is surjective statement
- First isomorphism theorem for linear maps statement
- Second isomorphism theorem for vector spaces statement
- Third isomorphism theorem for vector spaces statement
- Universal property of the quotient space statement
- Space of linear maps definition
- Dimension of the space of linear maps statement
- Endomorphism definition
- Algebra of endomorphisms object
- Projection definition
- Matrix of a linear map definition
- Linear map of a matrix definition
- Multiplication by a matrix as a linear map definition
- Matrix of a composition is the product of matrices statement
- Column space definition
- Row space definition
- Null space definition
- Rank of a matrix definition
- Row rank equals column rank statement
- Rank of a product of matrices statement
- Change of basis matrix definition
- Change of basis formula statement
- Similar matrices definition
- Equivalent matrices definition
- Trace of a linear map definition
- Trace of a linear map is the trace of its matrix statement
- Linear form definition
- Dual space definition
- Dual basis definition
- Dimension of the dual space statement
- Canonical isomorphism with the double dual statement
- Annihilator of a subspace definition
- Dimension of the annihilator statement
- Dual map definition
- Matrix of the dual map is the transpose statement
- Multilinear map definition
- Alternating multilinear map definition
- Determinant definition
- Leibniz formula for determinants statement
- Determinant of a 2 by 2 matrix statement
- Rule of Sarrus statement
- Determinant is multilinear in the rows statement
- Determinant of a matrix with two equal rows statement
- Determinant of a matrix with a zero row statement
- Scaling a row scales the determinant statement
- Adding a multiple of one row to another preserves the determinant statement
- Permuting rows multiplies the determinant by the sign statement
- Determinant of the identity matrix statement
- Determinant of a triangular matrix statement
- Determinant of a transpose statement
- Determinant is multiplicative statement
- Determinant of a scalar multiple statement
- Determinant of the inverse statement
- A matrix is invertible iff its determinant is a unit statement
- Determinant of a block triangular matrix statement
- Minor definition
- Cofactor definition
- Laplace expansion along a row statement
- Laplace expansion along a column statement
- Adjugate matrix definition
- Product of a matrix and its adjugate statement
- Inverse in terms of the adjugate statement
- Cramer's rule statement
- Vandermonde determinant statement
- Cauchy–Binet formula statement
- Determinant via the Schur complement statement
- Sylvester's determinant theorem statement
- Matrix determinant lemma statement
- Determinant of a linear map definition
- Determinant of a linear map is the determinant of its matrix statement
- Determinant of a family of vectors in a basis definition
- Determinant as signed volume concept
- Special linear group object
- Orientation of a real vector space definition
- Eigenvalue definition
- Eigenvector definition
- Eigenspace definition
- Spectrum of a matrix definition
- Characteristic polynomial definition
- Characteristic polynomial of a linear map definition
- Characteristic polynomial does not depend on the basis statement
- Eigenvalues are the roots of the characteristic polynomial statement
- Trace is a coefficient of the characteristic polynomial statement
- Determinant is a coefficient of the characteristic polynomial statement
- Trace is the sum of the eigenvalues statement
- Determinant is the product of the eigenvalues statement
- Eigenvectors with distinct eigenvalues are linearly independent statement
- Algebraic multiplicity definition
- Geometric multiplicity definition
- Existence of an eigenvalue over an algebraically closed field statement
- Gershgorin circle theorem statement
- Polynomial of an endomorphism definition
- Annihilating polynomial definition
- minimal polynomial definition
- Minimal polynomial divides every annihilating polynomial statement
- Eigenvalues are the roots of the minimal polynomial statement
- Cayley–Hamilton theorem statement
- Cayley–Hamilton theorem for linear maps statement
- Minimal polynomial divides the characteristic polynomial statement
- Diagonalisable matrix definition
- A matrix is diagonalisable iff it has a basis of eigenvectors statement
- Distinct eigenvalues imply diagonalisable statement
- Diagonalisable iff the minimal polynomial has distinct roots statement
- Powers of a diagonalisable matrix concept
- Simultaneous diagonalisation statement
- Invariant subspace definition
- Restriction of an endomorphism to an invariant subspace definition
- Triangularisation statement
- Generalised eigenspace definition
- Generalised eigenvector definition
- Decomposition into generalised eigenspaces statement
- Nilpotent endomorphism definition
- Jordan block definition
- Jordan normal form statement
- Semisimple endomorphism definition
- Jordan–Chevalley decomposition statement
- Companion matrix definition
- Rational canonical form statement
- Matrix exponential definition
- Exponential of a sum of commuting matrices statement
- Determinant of a matrix exponential statement
- Dot product definition
- Inner product definition
- Inner product space definition
- Norm definition
- Normed vector space definition
- Norm induced by an inner product definition
- Cauchy–Schwarz inequality statement
- Triangle inequality for norms statement
- Parallelogram law statement
- Polarisation identity statement
- Orthogonal vectors definition
- Pythagorean theorem in an inner product space statement
- Orthonormal family definition
- An orthonormal family is linearly independent statement
- Orthonormal basis definition
- Gram–Schmidt process concept
- Gram–Schmidt vectors are orthogonal statement
- Existence of an orthonormal basis statement
- Coordinates in an orthonormal basis statement
- Bessel's inequality statement
- Parseval's identity statement
- Orthogonal complement definition
- A space is the direct sum of a subspace and its orthogonal complement statement
- Dimension of the orthogonal complement statement
- Orthogonal complement of the orthogonal complement statement
- Orthogonal projection definition
- Orthogonal projection is the closest point statement
- Least squares concept
- Gram matrix definition
- Riesz representation theorem in finite dimensions statement
- Adjoint of a linear map definition
- Matrix of the adjoint is the conjugate transpose statement
- Self-adjoint operator definition
- Eigenvalues of a self-adjoint operator are real statement
- Eigenspaces of a self-adjoint operator are orthogonal statement
- Spectral theorem statement
- Spectral theorem for Hermitian matrices statement
- Rayleigh quotient definition
- Normal operator definition
- Normal matrix definition
- Spectral theorem for normal matrices statement
- Linear isometry definition
- A linear isometry preserves the inner product statement
- Orthogonal matrix definition
- Orthogonal group object
- Special orthogonal group object
- Unitary matrix definition
- Unitary group object
- Rotation matrix example
- Reflection definition
- Cartan–Dieudonné theorem statement
- QR decomposition statement
- Positive definite matrix definition
- Positive semidefinite matrix definition
- Eigenvalues of a positive definite matrix are positive statement
- Gram matrices are positive semidefinite statement
- Sylvester's criterion statement
- Cholesky decomposition statement
- Square root of a positive semidefinite matrix definition
- Singular value definition
- Singular value decomposition statement
- Polar decomposition statement
- Moore–Penrose inverse definition
- Bilinear form definition
- Matrix of a bilinear form definition
- Change of basis for a bilinear form statement
- Symmetric bilinear form definition
- Alternating bilinear form definition
- Nondegenerate bilinear form definition
- Orthogonal family for a bilinear form definition
- Sesquilinear form definition
- Quadratic form definition
- Polar form of a quadratic form definition
- Symmetric bilinear form associated with a quadratic form definition
- Diagonalisation of a quadratic form statement
- Sylvester's law of inertia statement
- Signature of a quadratic form definition
- Classification of complex quadratic forms statement
- positive definite quadratic form definition
- Principal axis theorem statement
- Tensor product of vector spaces definition
- Universal property of the tensor product statement
- Basis of a tensor product statement
- Dimension of a tensor product statement
- Kronecker product definition
- Exterior algebra definition
- Module over a ring definition
- Submodule definition
- Free module definition
- Finitely generated module definition