Linear Algebra/Index

A accuracy of Gauss' method addition vector additive inverse adjoint matrix angle antipodal antisymmetric matrix argument Arithmetic-Geometric Mean Inequality arrow diagram 1, 2, 3, 4, 5 augmented matrix automorphism dilation reflection rotation B back-substitution base step of induction basis 1, 2, 3 change of definition natural orthogonal orthogonalization orthonormal standard 1, 2 standard over the complex numbers string best fit line binary relation block matrix box orientation sense volume C C language classes equivalence canonical form for row equivalence for matrix equivalence for nilpotent matrices for similarity canonical representative Cauchy-Schwarz Inequality Caley-Hamilton theorem change of basis characteristic equation polynomial value vector characterized Chemistry problem 1, 2, 3 central projection circuits parallel series series-parallel closure of rangespace of nullspace codomain cofactor column vector column rank full column space complement complementary subspaces orthogonal complex numbers vector space over component composition self computer algebra systems concatenation condition number congruent figures congruent plane figures contradiction contrapostivite convex set coordinates homogeneous with respect to a basis corollary correspondence 1, 2 cosets Cramer's Rule cross product crystals diamond graphite salt unit cell D da Vinci, Leonardo determinant 1, 2 cofactor Cramer's Rule definition exists 1, 2, 3 Laplace Expansion minor Vandermonde permutation expansion 1, 2 diagonal matrix 1, 2 diagonalizable difference equation homogeneous dilation matrix representation dimension physical dilation 1, 2 direct map direct sum definition of two subspaces external internal direction vector distance-preserving division theorem domain dot product double precision dual space E echelon form leading variable free variable reduced eigenvalue of a matrix of a transformation eigenvector of a matrix of a transformation eigenspace element elementary matrix elementary reduction matrices elementary reduction operations pivoting rescaling swapping elementary row operations empty Erlanger Program entry equivalence class canonical representative representitive equivalence relation 1, 2 row equivalence isomorphism matrix equivalence matrix similarity equivalent statements Euclid even functions 1, 2 even polynomials external direct sum F Fibonacci sequence field definition finite-dimensional vector space flat form free variable full column rank full row rank function 1, 2 argument codomain composition composition correspondence domain even identity inverse 1, 2 inverse image left inverse multilinear range restriction odd one-to-one function onto right inverse structure preserving 1, 2 see homomorphism two sided inverse value well-defined zero Fundamental Theorem of Linear Algebra G Gauss' Method accuracy back-substitution elementary operations Gauss-Jordan Gauss-Jordan Gaussian operations generalized nullspace generalized rangespace Gram-Schmidt Orthogonalization Geometry of Eigenvalues Geometry of Linear Maps H historyless Markov Chain homogeneous homogeneous coordinate vector homogeneous coordinates homorphism composition matrix representation 1, 2, 3 nonsingular 1, 2 nullity nullspace rank 1, 2 rangespace rank zero I ideal line ideal point identity function identity matrix 1, 2 identity function if-then statement ill-conditioned image under a function improper subspace incidence matrix index of nilpotentcy induction 1, 2 inductive step of induction inherited operations inner product Input-Output Analysis internal direct sum 1, 2 intersection invariant subspace definintion inverse additive left inverse matrix right inverse two-sided inverse function inverse image inversion isometry isomorphism 1, 2, 3 characterized by dimension definition of a space with itself J Jordon form represents similarity classes Jordon block K kernel Kirkoff's Laws Klein, F. L Laplace Expansion leading variable least squares lemma length Leontief, W. line at infinity in projective plane of best fit linear tranpose operation linear combination Linear Combination Lemma linear equation coefficients constant homogeneous solution of Cramer's Rule Gausses' Method Gauss-Jordan system of satisfied by a vector linear map dilation see homomorphism reflection rotation 1, 2 skew trace linear recurrence linear relationship linear surface linear transformation linearly dependent linearly independent LINPACK M map extended linearly distance-preserving self composition Maple Markov Chain historyless Markov matrix material implication Mathematica mathematical induction1, 2 MATLAB matrix adjoint antisymmetric augmented block 1, 2 change of basis characteristic polynomial cofactor column column space condition number determinant 1, 2 diagonal matrix 1, 2 diagonalizable diagonalized eigenvalue eigenvector elementary reduction 1, :2 entry equivalent form identity 1, 2 incidence inverse 1, 2 existence left inverse main diagonal Markov minimal polynomial minor nilpotent nonsingular orthogonal orthonormal right inverse scalar multiple skew-symmetric similar similarity singular submatrix sum symmetric 1, 2, 3,4, 5, 6 trace 1, 2, 3 transition transpose 1, 2, 3 Markov matrix-vector product minimal polynomial multiplication nonsingular 1, 2 permutation principle diagonal rank representation row row equivalence row rank row space scalar multiple singular sum symmetric 1, 2, 3,4, 5 trace 1, 2 transpose 1, 2, 3 triangular 1, 2, 3 unit Vandermonde matrix equivalence definition cononical form rank characterization mean arithmetic geometric member method of powers minimal polynomial 1, 2 minor morphism multilinear multiplication matrix-matrix matrix-vector MuPAD mutual inclusion 1, 2 N natural representative networks Kirkoff's Laws nilpotent canonical form for definition matrix transformation nilpotentcy index nonsingular homomorphism matrix normalize nullity nullspace closure of generalized O Octave odd functions 1, 2 odd polynomials one-to-one function onto function opposite map order of a recurrence ordered pair orientation 1, 2 orthogonal basis complement mutually projection matrix orthogonalization orthonormal basis orthonormal matrix P pair ordered parallelepiped parallelogram rule parameter partition matrix equivalence classes partial pivoting partitions row equivalence classes isomorphism classes matrix equivalence classes Pascal's triangle permutation inversions matrix signum permutation expansion 1, 2 perp perpendicular permutation expansion perspective triangles Physics problem pivoting on rows full partial scaled plane figure congruence polynomial division theorem even odd of a map of a matrix minimal point at infinity in the projective plane populations, stable potential powers, method of preserves structure probability vector projection 1, 2, 3, 4 along a subspace central vanishing point onto a line onto a subspace orthogonal 1, 2 Projective Geometry projective plane Duality Principle ideal line ideal points lines proof techniques induction proper subspace subset propositions equivalent Q quantifier 1, 2 existential universal R range rangespace closure of generalized rank 1, 2 column of a homomorphism 1, 2 row recurrence 1, 2 homogeneous initial conditions reduced echelon form reflection glide reflection about a line matrix representation reflexivity relation equivalence reflexive symmetric transitive relationship linear representation of a vector of a matrix representative canonical for row equivalence of matrix equivalence classes of similarity classes rescaling rows resistance equivalent resistor restriction rigid motion rotation 1, 2, 3 matrix representation rounding error row vector row equivalence row rank full row space S scaled partial pivoting scalar scalar multiple matrix vector scalar multiplication vector 1, 2 scalar product Schur's triangularization lemma Schwarz Inequality SciLab self composition of maps sense sequence concatenation set complement element empty empty intersection linearly dependent linearly independent member mutual inclusion 1, :2 proper subset span of subset union signum similar 1, 2, 3 canonical form similar triangles similarity transformation single precision singular matrix size 1, 2 sgn see signum skew skew-symmetric span of a singleton spin square root stable populations standard basis state absorbing Statics problem string basis of basis vectors structure preservation submatrix subset subspace closed complementary direct sum definition improper independence invariant orthogonal complement proper sum sum of matrices of subspaces vector 1, 2, 3, 4 summation notation for permutation expansion swapping rows symmetric matrix 1, 2, 3, 4 symmetry system of linear equations Gauss' Method solving T theorem trace 1, 2, 3 transformation characteristic polynomial composed with itself diagonalizable eigenvalue eigenvector eigenspace Jordon form minimal polynomial nilpotent canonical representative projection size change transition matrix transitivity translation transpose 1, 2 interaction with sum and scalar multiplication determinant 1, 2 triangles similar Triangle Inequality triangular matrix triangularization trivial space 1, 2 turning map matrix representation U union unit matrix V vacuously true value Vandermonde determinant matrix vanishing point vector 1, 2 angle between vecots canonical position closure column complex scalars component cross product direction dot product free homogeneous coordinate length natural position orthogonal probability representation of 1, 2 row satisfies an equation scalar multiple scalar multiplication 1, 2 sum 1, 2, 3, 4 unit zero vector 1, 2 vector space basis definition 1, 2 dimension dual finite-dimensional homomorphism isomorphism map over complex numbers subspace trivial 1, 2 Venn diagram voltage drop volume voting paradox: majority cycle rational preference spin voting paradoxes W Wheatstone bridge well-defined X Y Z zero divisor 1, 2 zero homomorphism zero vector 1, 2