Linear algebra problems

244 problems · Matrices, vector spaces, linear maps, determinants and diagonalisation.

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Ex.13Checking linearity

#linear map #definition #examples

Linear maps

High school

Ex.14Matrix addition and scalar multiplication

#matrix operations #addition #scalar multiplication

Matrices and linear systems

High school

Ex.35Eigenspaces are subspaces

#eigenspace #subspace #linear map

Eigenvalues and diagonalisation

Ex.85Matrix of a linear map

#matrix representation #canonical basis #linear map

Linear maps

High school

Ex.108Computing a kernel

#kernel #linear map #basis

Linear maps

Ex.118Basis of a kernel

#kernel #basis #linear map

Linear maps

Ex.120Computing an image

#image #linear map #generators

Linear maps

Ex.124Rank of a linear map

#rank #image dimension #matrix

Linear maps

Ex.126Rank-nullity theorem on an example

#rank-nullity #rank #kernel dimension

Linear maps

Ex.128An injective linear map

#injectivity #kernel #linear map

Linear maps

Ex.130A surjective linear map

#surjectivity #rank #linear map

Linear maps

Ex.131An isomorphism

#isomorphism #inverse #linear map

Linear maps

Ex.133Similar matrices have the same eigenvalues

#similarity #eigenvalues #characteristic polynomial

Eigenvalues and diagonalisation

Ex.134Inverse of an endomorphism

#inverse #endomorphism #matrix inverse

Linear maps

Ex.136Composition and matrix product

#composition #matrix product #representation

Linear maps

Ex.138Composition of linear maps

#composition #matrix product #linear maps

Linear maps

Ex.140Kernel of a composition

#composition #kernel #inclusion

Linear maps

Ex.143Image of a composition

#composition #image #inclusion

Linear maps

Ex.145Rank inequality for a composition

#rank #composition #inequality

Linear maps

Ex.146Rank is invariant under invertible multiplications

#rank #invertible matrices #invariance

Linear maps

Ex.148A matrix with a single eigenvalue

#eigenvalues #eigenspace #diagonalization #Jordan block

Eigenvalues and diagonalisation

Ex.150Change of basis for a linear map

#change of basis #matrix representation #similarity

Linear maps

Ex.151Invariants under similarity

#similarity #trace #determinant #rank #characteristic polynomial

Linear maps

Ex.153A linear projection

#projection #idempotence #kernel and image

Linear maps

Ex.155A linear symmetry

#symmetry #involution #linear map

Linear maps

Ex.157A nilpotent endomorphism

#nilpotent #kernel #image

Linear maps

Ex.160Kernel and image of an endomorphism - when are they supplementary?

#kernel #image #direct sum #endomorphism

Linear maps

Ex.162Projection onto a subspace

#orthogonal projection #linear map #kernel and image

Linear maps

Ex.164Trigonalization and eigenvalues

#trigonalization #characteristic polynomial #real eigenvalues

Eigenvalues and diagonalisation

Ex.165Kernel and image of a matrix and its transposition

#kernel #image #transpose #orthogonal complement

Linear maps

Ex.166Eigenvectors and injectivity

#eigenvectors #injectivity #kernel

Linear maps

Ex.169Diagonalizability and the sum of eigenspaces

#diagonalization #eigenspaces #dimension

Linear maps

Ex.170Invariant subspaces

#invariant subspace #endomorphism #restriction

Linear maps

Ex.173Induced map on a quotient

#quotient space #induced map #well-defined

Linear maps

Ex.174Functions of a diagonalizable matrix

#functions of matrices #diagonalization #polynomial #well-definedness

Eigenvalues and diagonalisation

Ex.189Inverse of a 2×2 matrix

#inverse #determinants

Matrices and linear systems

High school

Ex.231Determinant of a 2×2 matrix

#determinant #2x2 matrices #invertibility

Determinants

High school

Ex.246Inner product axioms

#inner product #axioms #dot product

Inner product spaces

High school

Ex.266Definition and matrix of a quadratic form

#quadratic form #symmetric matrix #definition

Quadratic forms

High school

Ex.285Checking the vector space axioms

#vector space axioms #examples #counterexamples

Vector spaces and subspaces

High school

Ex.319A plane as a subspace

#subspaces #basis

Vector spaces and subspaces

Ex.327Inverting I − A

#matrices #nilpotent

Matrices and linear systems

Ex.328Kernel of a matrix

#kernel #rank

Matrices and linear systems

Ex.449Determinant of a 3×3 matrix by Sarrus' rule

#determinant #Sarrus rule #3x3 matrices

Determinants

High school

Ex.450Determinant of an inverse

#inverse #determinant #formula

Determinants

Ex.451Effect of row operations on the determinant

#row operations #determinant properties #multilinearity

Determinants

High school

Ex.452Determinant of a triangular matrix

#triangular matrix #determinant #diagonal

Determinants

High school

Ex.460Eigenvalues and eigenvectors of a 2×2 matrix

#eigenvalues #eigenvectors #characteristic polynomial

Eigenvalues and diagonalisation

High school

Ex.461A 2×2 matrix with distinct eigenvalues

#eigenvalues #eigenvectors #distinct roots

Eigenvalues and diagonalisation

High school

Ex.491Cauchy-Schwarz inequality

#Cauchy-Schwarz #inequality #inner product

Inner product spaces

High school

Ex.492Orthogonal families are independent

#orthogonal family #linear independence #proof

Inner product spaces

High school

Ex.506Determinant of a 3×3 matrix by Sarrus' rule

#determinant #Sarrus rule #3x3 matrices

Matrices and linear systems

High school

Ex.507Inverse of a 2×2 matrix

#inverse #determinant #2x2 matrices

Matrices and linear systems

Ex.508Matrix multiplication

#matrix multiplication #non-commutativity

Matrices and linear systems

High school

Ex.509Powers of a diagonal matrix

#diagonal matrix #powers #induction

Matrices and linear systems

Ex.510Properties of the trace

#trace #cyclic property #similarity invariance

Matrices and linear systems

Ex.511Properties of the transpose

#transpose #properties #matrix operations

Matrices and linear systems

Ex.557Polar form of a quadratic form

#polar form #symmetric bilinear form #quadratic form

Quadratic forms

High school

Ex.590Recognizing bases

#basis #dimension #canonical basis

Vector spaces and subspaces

High school

Ex.594Linear independence

#linear independence #definition #examples

Vector spaces and subspaces

High school

Ex.595Span of a set of vectors

#span #linear combinations #generating set

Vector spaces and subspaces

High school

Ex.596The subspace criterion

#subspace criterion #examples

Vector spaces and subspaces

High school

Ex.597Which subsets are subspaces?

#subspace criterion #counterexamples

Vector spaces and subspaces

High school

Ex.697Powers of a matrix

#diagonalisation #matrices

Eigenvalues and diagonalisation

Ex.704Powers of a 2×2 matrix with eigenvalues 1 and 2

#diagonalisation #matrices

Eigenvalues and diagonalisation

Ex.896The determinant is alternating

#alternating #equal columns #sign change

Determinants

Ex.897Determinant of a block triangular matrix

#block matrices #determinant #triangular

Determinants

Ex.898A trigonometric determinant

#trigonometric determinant #cos #sin #identity

Determinants

Ex.899Cramer's rule for a 2×2 system

#Cramer's rule #linear system #determinant

Determinants

High school

Ex.900Determinant of a cyclic permutation matrix

#permutation matrix #cyclic #determinant

Determinants

Ex.901Determinant of a diagonalisable matrix

#diagonalisation #determinant #eigenvalues

Determinants

Ex.902Expansion along a column

#cofactor expansion #minors #determinant

Determinants

High school

Ex.903Expansion along a row

#cofactor expansion #minors #determinant

Determinants

High school

Ex.904Multilinearity of the determinant

#multilinearity #determinant properties #proof

Determinants

Ex.905Determinant of an orthogonal matrix

#orthogonal matrix #determinant #isometry

Determinants

Ex.906Determinant and rank

#determinant #rank #invertibility

Determinants

Ex.907Determinant of the transpose

#transpose #determinant #proof

Determinants

Ex.933A 2×2 matrix with a double eigenvalue

#eigenvalues #double root #diagonalizability

Eigenvalues and diagonalisation

High school

Ex.934A 3×3 matrix with distinct eigenvalues

#eigenvalues #eigenvectors #3x3 matrices

Eigenvalues and diagonalisation

Ex.935Eigenvalues of a symmetric matrix

#symmetric matrix #eigenvalues #trace and determinant

Eigenvalues and diagonalisation

Ex.936The characteristic polynomial

#characteristic polynomial #eigenvalues #trace #determinant

Eigenvalues and diagonalisation

Ex.937Eigenvalues of a rotation

#rotation #complex eigenvalues #2x2 matrices

Eigenvalues and diagonalisation

Ex.938Similar matrices have the same eigenvalues

#similarity #eigenvalues #characteristic polynomial

Eigenvalues and diagonalisation

Ex.939Powers via diagonalization

#diagonalization #powers #matrix

Eigenvalues and diagonalisation

Ex.940A basis of an eigenspace

#eigenspace #basis #dimension

Eigenvalues and diagonalisation

Ex.941Sum of eigenvalues equals trace

#eigenvalues #trace #characteristic polynomial

Eigenvalues and diagonalisation

Ex.942Trace and determinant of a 3×3 matrix via eigenvalues

#trace #determinant #eigenvalues

Eigenvalues and diagonalisation

Ex.1021Gram-Schmidt orthonormalization

#Gram-Schmidt #orthonormal basis #algorithm

Inner product spaces

Ex.1022An orthogonal basis from the standard basis

#orthogonal basis #Gram-Schmidt #polynomial space

Inner product spaces

Ex.1023Orthogonal complement

#orthogonal complement #dimension #direct sum

Inner product spaces

Ex.1024Orthogonal matrices and isometries

#orthogonal matrix #isometry #inner product preservation

Inner product spaces

Ex.1025Triangle inequality

#triangle inequality #norm #inner product

Inner product spaces

High school

Ex.1048Block multiplication

#block matrices #multiplication #structure

Matrices and linear systems

Ex.1049Change of basis

#change of basis #transition matrix #similarity

Matrices and linear systems

Ex.1050Cramer's rule

#Cramer's rule #determinant #linear system

Matrices and linear systems

Ex.1051Determinant of a block triangular matrix

#block matrices #determinant #triangular structure

Matrices and linear systems

Ex.1052Using determinant properties

#determinant #row operations #properties

Matrices and linear systems

Ex.1053Eigenvalues and eigenvectors of a 2×2 matrix

#eigenvalues #eigenvectors #characteristic polynomial

Matrices and linear systems

Ex.1054Solving a linear system by Gauss-Jordan

#linear system #Gauss-Jordan #row reduction

Matrices and linear systems

High school

Ex.1055Gram-Schmidt orthogonalization

#Gram-Schmidt #orthogonalization #orthonormal basis

Matrices and linear systems

Ex.1056Invertibility depending on a parameter

#invertibility #parameter #determinant

Matrices and linear systems

Ex.1057Kernel and image of a matrix

#kernel #image #basis

Matrices and linear systems

Ex.1058Rank-nullity theorem

#rank-nullity #kernel #image

Matrices and linear systems

Ex.1059Powers of a nilpotent matrix

#nilpotent #powers #induction

Matrices and linear systems

Ex.1060Computing the rank of a matrix

#rank #Gaussian elimination #row reduction

Matrices and linear systems

Ex.1061Similarity invariants

#similarity #characteristic polynomial #trace #determinant

Matrices and linear systems

Ex.1145Reducing a quadratic form by completing the square

#reduction #completing square #quadratic form

Quadratic forms

Ex.1146Negative definite quadratic form

#negative definite #signature #Sylvester

Quadratic forms

Ex.1147A degenerate quadratic form

#degenerate #signature #quadratic form

Quadratic forms

Ex.1148The hyperbola $x^2-y^2=1$

#hyperbola #quadratic form #geometry

Quadratic forms

Ex.1149An indefinite quadratic form

#indefinite #signature #quadratic form

Quadratic forms

Ex.1194A basis of a hyperplane in $\mathbb{R}^4$

#basis #hyperplane #dimension

Vector spaces and subspaces

Ex.1195Image of a basis under a linear map

#basis #image #linear map

Vector spaces and subspaces

Ex.1196A basis of a polynomial space

#polynomial spaces #basis #dimension

Vector spaces and subspaces

Ex.1197The canonical basis and matrix representation

#canonical basis #matrix representation #linear map

Vector spaces and subspaces

Ex.1199Eigenspaces as subspaces

#eigenspaces #subspaces #dimension

Vector spaces and subspaces

Ex.1200Dimension of a subspace

#dimension #subspace #equations

Vector spaces and subspaces

Ex.1205Independence depending on a parameter

#linear independence #parameter #determinant

Vector spaces and subspaces

Ex.1206Isomorphism criterion

#isomorphism #dimension #injective linear map

Vector spaces and subspaces

Ex.1207Basis of a kernel

#kernel #basis #linear map

Vector spaces and subspaces

Ex.1208Kernel and image of a linear map

#kernel #image #linear map #rank-nullity

Vector spaces and subspaces

Ex.1209Kernel of a differentiation-like map

#kernel #differentiation #polynomials

Vector spaces and subspaces

Ex.1216The space of linear maps

#linear maps #vector space structure #dimension

Vector spaces and subspaces

Ex.1217Injective linear map and kernel

#injectivity #kernel #linear map

Vector spaces and subspaces

Ex.1218Column space and row space

#column space #row space #rank

Vector spaces and subspaces

Ex.1219Kernel of a 3×3 matrix and basis

#kernel #basis #matrix #rank-nullity

Vector spaces and subspaces

Ex.1220Subspace generated by a family

#span #generated subspace #dimension

Vector spaces and subspaces

Ex.1221Subspace generated by columns of a matrix

#column space #generators #rank

Vector spaces and subspaces

Ex.1222Subspace inclusion and dimension

#inclusion #dimension #equality

Vector spaces and subspaces

Ex.1223The direct sum criterion

#direct sum #criterion #proof

Vector spaces and subspaces

Ex.1224Surjective linear map and image

#surjectivity #image #linear map

Vector spaces and subspaces

Ex.1631Computing the inverse with cofactors

#cofactor #comatrix #inverse

Determinants

Ex.1632The cofactor formula for the inverse

#cofactor #comatrix #inverse formula

Determinants

Ex.1633Determinant of a companion matrix

#companion matrix #determinant #characteristic polynomial

Determinants

Ex.1634Cramer's rule for a 3×3 system

#Cramer's rule #3x3 determinant #linear system

Determinants

Ex.1635A cyclic 3×3 determinant

#cyclic determinant #factorisation #roots of unity

Determinants

Ex.1636A mixed determinant exercise

#determinant #row operations #factorisation

Determinants

Ex.1637The determinant as a volume

#determinant #volume #parallelepiped

Determinants

Ex.1638A numerical 4×4 determinant

#4x4 determinant #row reduction #computation

Determinants

Ex.1639Invertibility depending on a parameter

#parameter #invertibility #determinant

Determinants

Ex.1640Determinant of a product

#determinant of product #multiplicativity #proof

Determinants

Ex.1641Determinant of a skew-symmetric matrix

#skew-symmetric #determinant #odd dimension

Determinants

Ex.1642Trace and determinant via eigenvalues

#characteristic polynomial #trace #determinant #eigenvalues

Determinants

Ex.1643The Vandermonde determinant

#Vandermonde #determinant #factorisation

Determinants

Ex.1658Functions of a diagonalizable matrix

#functions of matrices #diagonalization #polynomial

Eigenvalues and diagonalisation

Ex.1659The diagonalizability criterion

#diagonalizability #multiplicity #eigenspaces

Eigenvalues and diagonalisation

Ex.1660An involution is diagonalizable

#involution #diagonalization #eigenvalues

Eigenvalues and diagonalisation

Ex.1661A projection is diagonalizable

#projection #diagonalization #eigenvalues

Eigenvalues and diagonalisation

Ex.1662Diagonalizing a 2×2 matrix

#diagonalization #change of basis #eigenvalues

Eigenvalues and diagonalisation

Ex.1663Diagonalizing a 3×3 matrix

#diagonalization #eigenspaces #change of basis

Eigenvalues and diagonalisation

Ex.1664Geometric versus algebraic multiplicity

#multiplicities #eigenspace #diagonalization

Eigenvalues and diagonalisation

Ex.1665A matrix with a single eigenvalue

#eigenvalues #eigenspace #diagonalization

Eigenvalues and diagonalisation

Ex.1745Adjoint of a linear map

#adjoint #linear map #inner product

Inner product spaces

Ex.1746Bessel's inequality

#Bessel inequality #orthonormal family #projection

Inner product spaces

Ex.1747Distance to a subspace

#distance #orthogonal projection #minimization

Inner product spaces

Ex.1748Orthogonal projection onto a subspace

#orthogonal projection #subspace #orthonormal basis

Inner product spaces

Ex.1749Completing an orthonormal family

#orthonormal family #basis #completion

Inner product spaces

Ex.1750Parseval's identity

#Parseval identity #orthonormal basis #norm

Inner product spaces

Ex.1751Self-adjoint operators and symmetric matrices

#self-adjoint #symmetric matrix #eigenvalues

Inner product spaces

Ex.1762The Cayley-Hamilton theorem

#Cayley-Hamilton #characteristic polynomial #matrix powers

Matrices and linear systems

Ex.1763Cholesky factorization of a 2×2 matrix

#Cholesky #positive definite #factorization

Matrices and linear systems

Ex.1764Diagonalizing a 2×2 matrix

#diagonalization #eigenvalues #change of basis

Matrices and linear systems

Ex.1765Diagonalizing a 3×3 matrix

#diagonalization #eigenvalues #eigenvectors

Matrices and linear systems

Ex.1766When is a matrix diagonalizable?

#diagonalization #eigenspaces #multiplicity

Matrices and linear systems

Ex.1767Eigenvalues of a 3×3 matrix

#eigenvalues #characteristic polynomial #trace and determinant

Matrices and linear systems

Ex.1768Inverse of a 3×3 matrix by row reduction

#inverse #Gaussian elimination #3x3 matrices

Matrices and linear systems

Ex.1769A linear recurrence and matrix powers

#linear recurrence #diagonalization #matrix powers

Matrices and linear systems

Ex.1770LU decomposition

#LU decomposition #Gaussian elimination #factorization

Matrices and linear systems

Ex.1771The minimal polynomial of a matrix

#minimal polynomial #characteristic polynomial #diagonalizability

Matrices and linear systems

Ex.1772Orthogonal projection onto a subspace

#orthogonal projection #projection matrix #orthonormal basis

Matrices and linear systems

Ex.1773Positive definiteness of a symmetric matrix

#positive definite #Sylvester criterion #eigenvalues

Matrices and linear systems

Ex.1774Powers of a companion matrix

#companion matrix #powers #recurrence

Matrices and linear systems

Ex.1775Reducing a quadratic form

#quadratic form #reduction #completing the square

Matrices and linear systems

Ex.1776Signature of a quadratic form

#quadratic form #signature #diagonalization

Matrices and linear systems

Ex.1777Rank depending on a parameter

#rank #parameter #Gaussian elimination

Matrices and linear systems

Ex.1778A linear system with a parameter

#linear system #parameter #discussion

Matrices and linear systems

Ex.1844Canonical form of a quadratic form

#canonical form #Gauss reduction #signature

Quadratic forms

Ex.1845Change of variables in a quadratic form

#change of variables #quadratic form #matrix

Quadratic forms

Ex.1846Comparison of two quadratic forms

#comparison #positive definite #inequalities

Quadratic forms

Ex.1847The cone $x^2+y^2-z^2=0$

#cone #quadratic form #signature

Quadratic forms

Ex.1848Diagonalization by congruence

#congruence #diagonalization #quadratic form

Quadratic forms

Ex.1849Reduction of a quadratic form by eigenvalues

#eigenvalues #orthogonal diagonalization #quadratic form

Quadratic forms

Ex.1850The ellipse $x^2+xy+y^2=1$

#ellipse #quadratic form #geometry

Quadratic forms

Ex.1851Gram matrix of a quadratic form

#Gram matrix #quadratic form #polar form

Quadratic forms

Ex.1852Checking positivity of a quadratic form

#positive definite #positivity #quadratic form

Quadratic forms

Ex.1853Positive definiteness of a 2×2 parameter family

#parameter #positive definite #minors

Quadratic forms

Ex.1854Reduction of a quadratic form with cross terms

#reduction #completing square #cross terms

Quadratic forms

Ex.1855Signature of a quadratic form in 3 variables

#signature #reduction #quadratic form

Quadratic forms

Ex.1856Sylvester's criterion for positive definiteness

#Sylvester criterion #positive definite #minors

Quadratic forms

Ex.1896Completing a family into a basis

#basis #completion #incomplete basis theorem

Vector spaces and subspaces

Ex.1897Change of basis in a polynomial space

#change of basis #transition matrix #polynomials

Vector spaces and subspaces

Ex.1898Existence of a supplementary subspace

#supplementary subspace #basis extension #existence

Vector spaces and subspaces

Ex.1899Hyperplanes and linear forms

#hyperplane #linear form #kernel #dimension

Vector spaces and subspaces

Ex.1900Dimension of a quotient space

#quotient space #dimension #proof

Vector spaces and subspaces

Ex.1901Direct sum of subspaces

#direct sum #supplementary subspaces #dimension

Vector spaces and subspaces

Ex.1902Decomposing a vector in a direct sum

#direct sum #decomposition #projection

Vector spaces and subspaces

Ex.1906The dual basis

#dual space #dual basis #linear forms

Vector spaces and subspaces

Ex.1907Eigenspaces sum to the whole space

#eigenspaces #direct sum #diagonalization

Vector spaces and subspaces

Ex.1909Orthonormal basis of a subspace

#Gram-Schmidt #orthonormal basis #subspace

Vector spaces and subspaces

Ex.1915Intersection of two kernels

#intersection #kernel #dimension

Vector spaces and subspaces

Ex.1916Intersection and sum of subspaces

#intersection #sum #dimension formula

Vector spaces and subspaces

Ex.1920Matrix of a linear map in a given basis

#matrix representation #basis #linear map

Vector spaces and subspaces

Ex.1921Rank-nullity on a 3×4 matrix

#rank-nullity #rank #matrix

Vector spaces and subspaces

Ex.1922Orthogonal complement

#orthogonal complement #dimension #inner product

Vector spaces and subspaces

Ex.1923A subspace of polynomials with a root condition

#polynomial subspace #evaluation #dimension

Vector spaces and subspaces

Ex.1924Quotient vector space

#quotient space #cosets #dimension

Vector spaces and subspaces

Ex.1927A subspace depending on a parameter

#parameter #subspace #dimension

Vector spaces and subspaces

Ex.1928Sum of two planes in $\mathbb{R}^3$

#sum #intersection #planes

Vector spaces and subspaces

Ex.1929Symmetric and skew-symmetric matrices

#symmetric #skew-symmetric #direct sum

Vector spaces and subspaces

Ex.1930The subspace of traceless matrices

#trace #subspace #dimension #matrices

Vector spaces and subspaces

Ex.1931Kernel of the transpose

#transpose #kernel #orthogonal complement #rank

Vector spaces and subspaces

Ex.2118Gram determinant and Cauchy-Schwarz

#Gram determinant #Cauchy-Schwarz #inner product

Determinants

Ex.2119The general Vandermonde determinant

#Vandermonde #determinant #general formula

Determinants

Ex.2120Hadamard's inequality

#Hadamard inequality #determinant #column norms

Determinants

Ex.2121Sylvester's determinant identity

#Sylvester identity #block matrices #determinant

Determinants

Ex.2124The Cayley-Hamilton theorem

#Cayley-Hamilton #characteristic polynomial #matrix identity

Eigenvalues and diagonalisation

Ex.2125Eigenvalues of a circulant matrix

#circulant matrix #eigenvalues #roots of unity

Eigenvalues and diagonalisation

Ex.2126Powers of a companion matrix and Fibonacci numbers

#companion matrix #Fibonacci #diagonalization

Eigenvalues and diagonalisation

Ex.2127Diagonalizability depending on a parameter

#diagonalizability #parameter #eigenspaces

Eigenvalues and diagonalisation

Ex.2128Trigonalization and eigenvalues

#trigonalization #characteristic polynomial #real eigenvalues

Eigenvalues and diagonalisation

Ex.2129Diagonalizability and the minimal polynomial

#minimal polynomial #diagonalization #criterion

Eigenvalues and diagonalisation

Ex.2130Spectral decomposition of a symmetric matrix

#spectral decomposition #symmetric matrix #projection

Eigenvalues and diagonalisation

Ex.2131Orthogonal diagonalization of a symmetric matrix

#symmetric matrix #orthogonal diagonalization #spectral theorem

Eigenvalues and diagonalisation

Ex.2168Fourier coefficients and orthogonal projection

#Fourier coefficients #trigonometric basis #orthogonal projection

Inner product spaces

Ex.2169Least squares and normal equations

#least squares #normal equations #orthogonal projection

Inner product spaces

Ex.2170Legendre polynomials

#Legendre polynomials #orthogonal polynomials #inner product

Inner product spaces

Ex.2171Orthogonal projection onto a column space

#orthogonal projection #column space #normal equations

Inner product spaces

Ex.2172Riesz representation theorem (finite-dimensional)

#Riesz representation #linear forms #inner product

Inner product spaces

Ex.2178Diagonalizability depending on a parameter

#diagonalization #parameter #characteristic polynomial

Matrices and linear systems

Ex.2179QR factorization by Gram-Schmidt

#QR factorization #Gram-Schmidt #orthogonalization

Matrices and linear systems

Ex.2180Singular value decomposition of a 2×2 matrix

#SVD #singular values #orthogonal matrices

Matrices and linear systems

Ex.2181Orthogonal diagonalization of a symmetric matrix

#symmetric matrix #orthogonal diagonalization #spectral theorem

Matrices and linear systems

Ex.2205Bounding a quadratic form by eigenvalues

#eigenvalues #Rayleigh quotient #inequalities

Quadratic forms

Ex.2206An inequality via a positive definite form

#inequality #positive definite #quadratic form

Quadratic forms

Ex.2207Sylvester's law of inertia

#inertia law #signature #invariance

Quadratic forms

Ex.2208LDL decomposition of a symmetric matrix

#LDL decomposition #Sylvester #quadratic form

Quadratic forms

Ex.2209Maximizing a quadratic form on the sphere

#maximum #sphere #eigenvalues

Quadratic forms

Ex.2210Minimizing a quadratic form on the sphere

#minimum #sphere #eigenvalues

Quadratic forms

Ex.2211Orthogonal diagonalization of a quadratic form

#orthogonal diagonalization #eigenvalues #quadratic form

Quadratic forms

Ex.2212Positive definiteness depending on a parameter

#parameter #positive definite #Sylvester

Quadratic forms

Ex.2213The Rayleigh quotient

#Rayleigh quotient #eigenvalues #optimization

Quadratic forms

Ex.2214Signature depending on a parameter

#signature #parameter #quadratic form

Quadratic forms

Ex.2215When are two quadratic forms equivalent?

#equivalence #signature #congruence

Quadratic forms

Ex.2222Proving the dimension formula

#dimension formula #proof #intersection and sum

Vector spaces and subspaces