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vector spaces rotation linear systems vector unit square unifying theorem triangular matrix stopping procedures standard position spanning solution space similar matrix scalars scalar power sequences polar decomposition pivot theorem normal matrix multiplication minors matrix transformations matrix linear systems linear system linear operators linear independence linear independence invertibility inverse inner product space hyperplane homogeneous linear systems geometry Gaussian elimination fundamental spaces of a matrix ellipsoid elementary products eigenvectors differential equations diagonalization diagonalizability determinants data compression coordinate systems convergence consistency complex number addition zero vector zero transformation zero subspace zero solution zero matrix zero yaw wires wireframes weights weighted least squares solutions weighted least squares problems weighted Euclidean inner product weighted Euclidean weighted volts voltage visible space vertices vertex matrix vertex velocity vectors vectors vector equation vector equation vector component vector addition as vector addition variations variance variable vanishing point vanishing values of a function using upper triangular matrix upper limit of summation upper Hessenberg form upper Hessenberg decomposition upper Hessenberg unusual types unstable algorithms unit sphere unit circle unitary matrix unitary diagonalization unitary unitarily diagonalizable square complex unit uniqueness uniqueness unifying theorems related to unifying theorem unified field theory underdetermined linear systems undefined product undefined types two-way connections two-point vector equations two matrix trivial subspace trivial solutions trivial fixed points trivial fixed trivial triple products trigonometric polynomials trigonometric triangular triangle rule for vector addition triangle inequality transpose of a matrix transpose of a transpose translation translation translated subspaces translated transition matrix between transition matrix transition between transition transformation trace of a trace to through two points through three or more linear transformations three-dimensional graphics three-dimensional thermodynamics the plane theory vs. implementation theorems theorem of Pythagoras theorem the line the inner product terms terminal point of vector technology matrix technology systems systems system of differential equations system symmetry property symmetric matrix symmetric matrix symmetric superposition principle summation notation sum sum successive overrelaxation successive displacements subtraction subtraction substitution subspaces subspaces subspace submatrix subdiagonals string theory strictly upper triangular matrix strictly triangular matrix strictly lower triangular matrix strictly diagonally dominant square stochastic matrix stochastic steady-state vectors steady state static equilibrium state of the variable state of the dynamical system state of the state of particle system standard unit vectors standard unit standard matrix for T standard matrix standard forms standard basis standard squares fits by square root of a matrix square root square matrix square spot color method sphere spectral decomposition spectral sparse linear systems sparse spans space-time continuum solving multiple linear systems with solving linear systems by solving linear systems solving solutions using solutions to systems of linear equations solutions to systems of linear solutions to solution space of linear system solution space solution set size singular value expansion singular value decomposition SVD singular value decomposition singular value decomposition singular value singular matrix singular simultaneous displacements similarity invariants similarity similar signed elementary products signed elementary sigma notation shifted inverse power method shifted inverse shifted shear sets to basis see Jacobi iteration sectors second derivative test search set screen coordinates scaling matrix scaling scalar triple products scalar triple scalar multiplication in vector spaces scalar multiple sample variance sample mean sample saddle point saddle row vectors row-vector form vector notation row reduction row operations row equivalence row echelon form row-column rule dot product rule row-column rule row roundoff error roundoff rotations about rotation matrix rotated out of standard position rosettes roman fonts roll term rise right singular vectors right singular right-handed coordinate systems right distributive law right respect to pair resistors resistance representing linear operators with two representation relativity relative minimum relative maximum relative error relative regular reflections of R" about the reflections about lines through reflections reduced singular value expansion reduced singular value decomposition reduced row echelon form reduced recurrence relations rectangular coordinates reciprocal of a complex number reciprocal real vector spaces real part real matrix real inner product spaces real eigenvalues of symmetric real axis of complex plane real reactants rate of convergence range range quotient complex numbers quadratic surfaces quadrics quadratic form pure imaginary numbers pseudoinverse and solution pseudoinverse properties shared by projection theorem projections onto projections products from chemical reactions product of matrix productive open economies production vectors production product product process color method probability vectors probability principal axes theorem principal axes of an ellipse principal argument of a complex principal preimage power sources powers of transition matrix powers of a matrix powers of a matrix powers of a power series representation power sequence powers power method and rate positive semidefinite quadratic forms positive definite quadratic forms positive definite matrix positive definite portrait mode polynomials polarization identity polar form polar form polar point-normal equations point-normal equations point pixels pivots pivot positions pivot columns perspective projection perpendicular vectors perpendicular permutation matrix permutation percentage error percentage partitioning partitioned matrix partitioned particular solution of consistent linear partial pivoting parametric equations parametric equations parameters parallel vectors parallel processing parallelogram rule for vector addition parallelogram equation for vectors parallelogram parallel LRC circuits parallel lines/planes parallel pair of bases overdetermined vs. underdetermined outside demand vectors outside outputs outer products outer product rule orthonormal vectors orthonormal sets orthonormal bases using orthonormal bases orthonormal orthogonal vectors orthogonal similarity orthogonal projections using orthogonal projections onto lines orthogonal projection orthogonal operators orthogonal matrix orthogonal linear operators as orthogonality property of least squares orthogonal diagonalization problem orthogonal diagonalization orthogonal complement orthogonal change of variable orthogonal bases origin orientation of axis of rotation optimization operations on open sectors open economies open onto transformations onto lines through the origin onto general subspaces onto on real vector space one-way connections one-to-one transformations one-to-one correspondence one-to-one and onto from viewpoint one-to-one ohms numerical methods for obtaining numerical analysis number nullity n-tuples notation norm-preserving linear operators norm-preserving normal vectors normalization of vectors normalization normal norm norm nontrivial solutions nontrivial nonsquare matrix nonsingular matrix nonsingular nonhomogeneous linear systems nonhomogeneous nondiagonalization case nondiagonalizable matrix nondiagonalizable nodes nilpotent matrix nilpotent negative semidefinite quadratic forms negative semidefinite negative definite quadratic forms negative negative natural isomorphisms natural multiplicity multiplicative inverse multiplication of vectors by multiplication modulus modulus minor axis minimum distance problems minimum distance problem minimum method of successive overrelaxation method of successive displacements method of simultaneous displacements mean square error mean square meanhead mean maximum entry scaling maximum matrix transformation matrix theory matrix spaces matrix representations matrix products as matrix polynomials matrix outer products matrix outer matrix operators matrix of linear representation with matrix of linear operator with respect matrix notation matrix inversion by matrix inversion matrix inner products matrix inner matrix for the matrix and scalar matrix matrix matrix with rows or columns that matrix as representations matrix mapping mantissa major vs. minor major axis main diagonal magnitude of a vector magnitude lower triangular matrix lower limit of summation long-term behavior lines through two lines through the origin lines/planes parallel to linear transformation with respect to linear transformations linear transformation linear systems with triangular linear operator with respect to basis linear manifolds linear isometry linear independence and homogeneous linear factorization linear equations linear dependence linear combinations of orthonormal linear combinations linear combination linear line length-preserving operators length length left singular vectors left singular left-handed coordinate systems left distributive law left least squares solution with respect to least squares problems least squares problem least squares leading variables leading numbers leading landscape mode junctions iterative methods italic fonts isomorphism involving three or more statements invertible matrix invertible invertibility invertibile linear operators inversion inverse transformations inverse power method inverse operations inverse of a matrix inverse of a inverse interpolating polynomials intermediate demand vectors intermediate demand intermediate integral inner product integral inputs/outputs inputs input-output analysis initial value problems initial point of vector initial hub vectors initial conditions initial authority vectors initial approximation infinite-dimensional vector spaces infinite-dimensional index of summation indefinite quadratic forms inconsistent linear systems inconsistent imaginary part of complex number imaginary axis of complex plane images ill-conditioned systems identity operators identity matrix identity hyperplanes passing through hyperplanes hyperbolas hub weights hubs hub homogeneous linear equations homogeneous first-order linear systems homogeneous first-order homogeneous equations homogeneous coordinates homogeneous higher-dimensional spaces higher-degree polynomials Hermitian matrix have all zeros gravity graph given geometry geometric mean geometric interpretation geometric genetics generated by A general solution general solution general relativity theory general linear transformations generalized vector generalized point generalized general inner product spaces general equation of a general fundamental theorem of algebra fundamental spaces of a matrix fundamental spaces fundamental solutions of systems fundamental solutions fundamental set of solutions function spaces functions function from transformation point of view free vector free variables free forward substitution forward phase Gauss-Jordan formula forms to force vector flops and cost of solving flops floating-point numbers flats fixed points of a fixed points fitting a curve to experimental first-order equations finite-precision arithmetic finite-precision finite-dimensional vector spaces finite-dimensional finding factor theorem factorization factoring extrema extrapolated extension of linearly independent expressions for determinant and traceo expressions exponential form of solution exponential exponential expansion existence exact arithmetic evaluation transformation evaluation difficulties for higher-order evaluation estimated relative error estimated percentage error error vectors error equivalent vectors equivalent statements equivalent equal vectors equivalent equilibrium equations with no equations equation equal equivalent vectors equal entries of a matrix entries empty graphs ellipses elimination elementary row operations on elementary row operations for solution elementary row/column operations elementary matrix elementary electrical potential electrical current electrical circuits eigenvectors/eigenvalues eigenvalues and eigenvectors eigenvalues eigenvalues eigenvalue decomposition EVD eigenvalue analysis eigenspaces eigenheads effect of changing bases on matrix edges economy dynamical systems drop double perp theorem dot product rule row-column rule dot-product-preserving operators dot product dot product dot dominant eigenvectors dominant eigenvalue dominant domains division of complex numbers division distributive law distances distance from a point to a distance between distance displacement discrete averaging model direct proportionality direction direction directed graphs directed dimension theorem for subspaces dimension theorem for subspaces dimension theorem for matrix dimension theorem for homogeneous dimension theorem dimension dimension dilations differential equations differentiable function difference of matrix difference diagonal matrix diagonalization of a diagonalizable operators diagonalizable matrix diagonalizable diagonal determining whether a vector is in a determinant of a determinant evalution by determinant demand vectors degenerate parallelograms degenerate conics defintion decomposition data compression data curves current cross product terms cross products cross critical points critical cost estimates for solving large coordinates with respect to coordinates coordinate planes about coordinate planes coordinate maps coordinate axes converse of a theorem converse contrapositive of a theorem contrapositive form contraction consumption vectors consumption matrix consumption constraints constrained extremum theorem constrained extremum problems constant of proportionality consistent linear systems consistent linear system consistent consistency consequences connectivity matrix connectivity graphs connectivity connections conjugate transpose conjugate conic sections conics conic sections conforming matrix sizes condition number condensation computing length computer roundoff error computer roundoff computer graphics compression compositions components components complex vector spaces complex plane complex number system complex numbers complex n-tuples complex n-spaces complex inner product space complex inner product complex exponentials complex Euclidean inner product complex Euclidean inner product complex Euclidean complex eigenvectors complex eigenvalues of real matrix complex eigenvalues complex dot product complex conjugate complete reactions complete linear factorization complete linear companion matrix companion commutative law comma-delimited form vector notation column vectors column-vector form vector notation column rule for matrix multiplication column-row rule column-row factorization column-row expansion column operations column and row rules for matrix column collinear vectors collinear cofactors cofactor expansions coefficients in linear combination coefficient matrix and column coefficient matrix coefficient matrix coefficient codomain closed under scalar multiplication closed under scalar closed under linear combinations closed under addition closed loops closed classification of conic sections using classification circuits circle choosing an algorithm for solving chemical formulas characteristic polynomials characteristic equations characteristic changing change of variable change of variable change central quadrics central ellipsoid central center of scaling center cancellation law branches bound vectors bound bilinear form between bases best mean square approximation best approximation problem best approximation basis vectors balanced chemical reactions axioms authority weights authority authorities augmented matrix augmented associative law associated with associated homogeneous linear systems associated arithmetic mean arithmetic argument argument applications applications application a plane antisymmetry property anticommutative matrix anticommutative angle-preserving operators angle between vectors angle between alternation alleles a linear operator a line algorithms algorithm for finding algorithm algebraic properties algebraic multiplicity algebraic and geometric multiplicity algebraic affine spaces adjoint of a matrix adjoint adjacency matrix adjacency additivity addition absolute value