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7.5: Index

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    70221
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    Index Index
    \(\cB\)-coordinates
    change of basis matrix, Subsection
    computing
    row reduction, Section
    with respect to an orthogonal basis, Subsection
    definition of, Definition
    informally, Paragraph
    labeling points, Section
    nonstandard grid, Section
    Algebraic multiplicity
    and diagonalizability, Theorem
    and geometric multiplicity, Theorem
    definition of, Definition
    equals one, Subsection
    of similar matrices, Theorem
    Approximate solution. See Least-squares
    Augmented matrix. See Matrix
    Basis
    and orthogonal projection, Corollary
    basis theorem, Theorem
    coordinates with respect to see \(\cB\)-coordinates
    definition of, Subsection
    infinitely many, Subsection
    making orthogonal, Theorem
    of \(\mathbb{R}^n \), Example Subsection
    of a column space, Theorem
    of a null space, Theorem
    of a span, Paragraphs
    span of an orthogonal set, Fact
    uniqueness with respect to, Fact
    Best-fit problem, Subsection
    best-fit ellipse, Example
    best-fit line, Subsection
    best-fit linear equation, Example
    best-fit parabola, Example
    best-fit trigonometric function, Example
    general setup, Paragraph
    Block Diagonalization Theorem, Theorem
    Characteristic polynomial
    and eigenvalues, Theorem
    definition of, Definition
    factoring by hand, Paragraphs
    form of, Theorem
    of a \(2\times 2\) matrix, Paragraphs
    of similar matrices, Theorem
    Codomain. See Transformation
    Cofactor. See Matrix
    Color space, Example
    Column rank. See Rank
    Column Space
    orthogonal complement of, Proposition
    Column space
    and invertibility, Theorem
    and rank see Rank
    basis of see Basis
    definition of, Definition
    is a subspace, Paragraph
    is row space of transpose, Paragraph
    of an orthogonal projection, Proposition
    orthogonal complement of, Subsection
    range of a transformation, Subsection
    versus the solution set, Subsection
    Column span. See Column space
    Complex conjugation. See Complex numbers
    Complex eigenvalue
    \(2\times 2\) matrices
    and rotation-scaling matrices, Theorem
    and rotation-scaling matrices, computing, Subsection
    different rotation-scaling matrices, Paragraph
    dynamics of, Note Example Example Example
    geometry of, Subsection
    \(3\times 3\) matrices
    dynamics of, Example
    conjugate pairs, Subsection
    existence of, Subsection
    Complex numbers
    absolute value, Item
    arithmetic of, Note
    conjugation, Item
    definition of, Definition
    real and imaginary parts of, Item
    vectors, Paragraph
    Consistent. See System of linear equations
    Cramer’s rule, Theorem
    and computing inverses, Theorem
    Determinant
    alternative defining properties of, Remark
    and column operations, Corollary
    and computing inverses, Theorem
    and powers of matrices, Corollary
    and row operations, Subsection
    and volumes, Theorem
    computation of
    cofactor expansion, Theorem
    row reduction, Subsection
    defining properties of, Subsection
    existence and uniqueness of, Theorem
    identity matrix, Subsection
    invertibility property, Proposition Theorem
    methods of computation, Subsection
    multilinearity property, Proposition
    multiplicativity property, Proposition
    and volumes, Remark
    of a \(2\times 2\) matrix, Definition Subsection Subsection
    of a \(3\times 3\) matrix, Subsection
    of similar matrices, Paragraph
    properties of, Subsection
    transpose property, Proposition
    Diagonal
    see Matrix, Definition
    Diagonalizability
    algebraic-geometric multiplicity criterion, Theorem
    criterion, Theorem
    criterion for similarity, Paragraphs
    definition of, Definition
    diagonal matrices, Subsection Example
    distinct eigenvalues, Subsection
    geometry of, Subsection
    is unrelated to invertibility, Subsection
    of \(2\times 2\) matrices, Subsection
    of \(3\times 3\) matrices, Example
    of a projection matrix, Proposition
    order of eigenvalues, Note
    powers of, Paragraphs
    projection, Example
    recipe, Subsection
    shear, Example
    similar matrices, Subsection
    worked example, Subsection
    Difference equation, Definition
    eigenvectors of, Example
    inverse of, Example Example
    Dimension
    definition of, Subsection
    of a column space, Definition
    of a null space, Definition
    of a solution set, Subsection Subsection
    of an orthogonal complement, Fact
    Domain. See Transformation
    Dot product
    and angles, Remark
    and distance, Definition
    and length, Fact
    definition of, Definition
    properties of, Note
    Eigenspace
    and the null space, Fact
    computation, Subsection
    definition of, Definition
    is a solution set, Paragraph
    is a subspace, Paragraph
    of a projection matrix, Proposition
    of similar matrices, Subsection
    orthogonal complement of, Example
    Eigenvalue
    algebraic multiplicity of see Algebraic multiplicity
    and diagonalizability, Subsection
    and invertibility, Theorem
    and stochastic matrices, Fact
    and the characteristic polynomial see Characteristic polynomial
    complex see Complex eigenvalue
    definition of, Subsection
    eigenvector for, Subsection
    geometric multiplicity of see Geometric multiplicity
    identity matrix, Example
    maximum number of, Subsection
    of a projection matrix, Proposition
    of a triangular matrix, Corollary
    of similar matrices, Subsection
    zero, Fact
    Eigenvector
    and collinearity, Paragraph
    and diagonalizability, Theorem
    and stochastic matrices, Definition
    computation, Subsection
    trick for \(2\times 2\) matrices, Note
    definition of, Subsection
    eigenvalue for, Subsection
    identity matrix, Example
    linear independence of, Fact
    of a projection matrix, Proposition
    of similar matrices, Theorem
    Elimination method, Subsection
    Ellipse
    area of see Volume
    best-fit problem, Example
    Equation of linear dependence. See Linear independence
    Free variable, Definition
    Function. See Transformation
    Fundamental theorem of algebra, Theorem
    Gaussian elimination. See Row reduction
    Geometric multiplicity
    and algebraic multiplicity, Theorem
    and diagonalizability, Theorem
    definition of, Definition
    of similar matrices, Theorem
    Google PageRank
    Google Matrix, Definition
    eigenvector of, Subsection
    importance matrix, Definition
    eigenvector of, Subsection
    importance rule, Definition
    Gram–Schmidt Process, Theorem
    detecting linear dependence, Subsection
    Homogeneous. See System of linear equations
    Identity matrix
    and identity transformation, Example
    and matrix multiplication, Note
    and standard coordinate vectors, Definition
    as a function, Example
    as a matrix transformation, Example
    definition of, Definition
    determinant of, Subsection
    eigenvalues of, Example
    eigenvectors of, Example
    similarity, Paragraph
    Identity transformation
    and composition, Note
    and identity matrix, Example
    definition of, Definition
    Imaginary number. See Complex numbers
    Imaginary part. See Complex numbers
    Implicit equation, Subsection Note
    Inconsistent. See System of linear equations
    Increasing span criterion. See Linear independence
    Inhomogeneous. See System of linear equations
    Invertible matrix
    and invertible transformation, Theorem
    basic facts, Fact
    computation
    \(2\times 2\) case, Proposition
    in general, Theorem
    using Cramer’ rule, Theorem
    definition of, Definition
    determinant of, Proposition Theorem
    inverse of, Item
    invertible matrix theorem, Theorem
    solving linear systems with, Theorem
    Invertible matrix theorem, Theorem Theorem
    Invertible transformation
    and invertible matrices, Theorem Theorem
    definition of, Definition
    one-to-one and onto, Proposition
    Least-squares
    and \(Ax=b_{\Col(A)}\), Paragraphs
    computation of
    complicated matrix formula, Theorem
    Projection Formula, Subsection
    row reduction, Theorem Subsection
    definition of, Definition
    picture of, Paragraphs
    uniqueness of, Theorem
    Line
    dimension-1 solution set, Subsection Subsection
    geometric definition of, Definition
    number line, Subsection
    orthogonal projection onto, Subsection Subsection
    parametric form of, Subsection
    Linear combination
    collinear vectors, picture of, Subsection
    definition of, Definition
    single vector, picture of, Subsection
    two vectors, picture of, Example
    Linear dependence. See Linear independence
    Linear dependence relation. See Linear independence
    Linear equation
    definition of, Definition
    system of see System of linear equations
    Linear Independence
    basic facts, Fact
    equation of linear dependence, Paragraph
    increasing span criterion, Theorem
    linear dependence relation, Paragraph
    pictures of, Subsection
    verifying, Subsection
    wide matrices, Subsection
    Linear independence
    and determinants, Corollary
    and invertibility, Theorem
    definition of, Subsection
    of an orthogonal set, Fact
    verifying
    with Gram–Schmidt, Subsection
    Linear transformation
    addition of see Transformation
    and volumes see Matrix transformation
    are matrix transformations, Subsection
    basic facts, Fact
    composition of see Transformation
    and matrix multiplication, Theorem
    linearity of, Theorem
    definition of, Definition
    dictionary, Note
    invertible see Invertible transformation
    scalar multiplication of see Transformation
    standard matrix of, Theorem
    orthogonal projection, Paragraph
    verifying nonlinearity, Example
    when defined by a formula, Paragraph
    Lower-triangular
    see Matrix, Definition
    Matrix
    addition of, Definition
    as a function Paragraph; Matrix transformation
    augmented, Paragraphs
    cofactor of, Definition
    and determinants, Theorem
    sign of, Paragraph
    definition of, Paragraphs
    determinant of see Determinant
    diagonal entries of, Definition
    inverse of see Invertible matrix
    invertible see Invertible matrix
    lower-triangular, Definition
    determinant of, Proposition
    eigenvalues of, Corollary
    minor of, Definition
    multiplication see Matrix multiplication
    nullity of see Nullity
    parallelepiped determined by, Theorem
    product with vector see Matrix-vector product
    projection see Orthogonal projection, standard matrix of
    rank of see Rank
    rotation-scaling see Rotation-scaling matrix
    scalar multiplication of, Definition
    similar see Similarity; Similarity
    size of, Subsection
    stochastic see Stochastic matrix
    trace of, Definition
    similar matrices, Paragraph
    transpose of, Definition
    and products, Fact
    determinant of, Proposition
    upper-triangular, Definition
    determinant of, Proposition
    eigenvalues of, Corollary
    Matrix equation
    always consistent, Theorem
    and invertibility, Theorem
    definition of, Definition
    equivalence with vector equation, Note
    solving with the inverse matrix, Theorem
    spans and consistency, Subsection
    Matrix multiplication
    and composition of transformations, Theorem
    and the matrix-vector product, Paragraph
    associativity of, Note
    caveats, Subsection
    definition of, Definition
    determinant of, Proposition
    inverse of, Item
    noncommutativity of, Example
    order of operations, Example
    powers, Paragraph
    and diagonalizability, Paragraphs
    and similarity, Fact Fact
    properties of, Note
    row-column rule, Paragraphs
    size of matrices, Subsection
    Matrix transformation
    addition of see Transformation
    and volumes, Theorem
    codomain of, Subsection
    composition of see Linear transformation; Transformation
    definition of, Definition
    dictionary, Note
    domain of, Subsection
    invertible see Invertible transformation
    linearity of, Paragraph
    of \(\mathbb{R}^2 \), Example
    one-to-one criteria, Theorem
    onto criteria, Theorem
    range of, Subsection
    scalar multiplication of see Transformation
    tall matrices, Note
    wide matrices, Note
    Matrix-vector product
    and matrix multiplication, Paragraph
    definition of, Definition
    row-column rule, Paragraphs
    with standard coordinate vectors, Fact
    Minor. See Matrix
    Multiplicity
    algebraic see Algebraic multiplicity
    geometric see Geometric multiplicity
    Nontrivial solution. See System of linear equations
    Null space
    and invertibility, Theorem
    basis of see Basis
    computing, Subsection
    definition of, Definition
    is a solution set, Subsection
    is a subspace, Subsection
    is the \(0\)-eigenspace, Fact
    of an orthogonal projection, Proposition
    orthogonal complement of, Subsection
    Nullity, Definition
    rank theorem, Theorem
    One-to-one
    and invertibility, Theorem
    criteria for matrix transformations, Theorem
    definition of, Definition
    equivalent formulations, Paragraph
    finding two vectors with the same image, Paragraph
    functions of one variable, Example
    negation of, Paragraph
    square matrices, Note
    versus onto, Paragraph
    wide matrices, Note
    Onto
    and invertibility, Theorem
    criteria for matrix transformations, Theorem
    definition of, Definition
    equivalent formulations, Paragraph
    finding a vector not in the range, Paragraph
    functions of one variable, Example
    negation of, Paragraph
    square matrices, Note
    tall matrices, Note
    versus one-to-one, Paragraph
    Orthogonal complement
    basic facts, Fact
    computation of, Subsection
    definition of, Definition
    dimension of, Fact
    of a column space, Proposition Subsection
    of a null space, Subsection
    of a row space, Subsection
    of a span, Paragraph Subsection
    of an eigenspace, Example
    orthogonal complement of, Fact
    pictures of, Paragraphs
    system of linear equations, Paragraph
    Orthogonal decomposition. See Orthogonal projection
    Orthogonal projection
    and \(\cB\)-coordinates, Subsection
    as a transformation, Subsection
    composed with itself, Proposition
    computation of
    complicated matrix formula, Corollary
    Projection Formula, Theorem
    row reduction, Theorem Subsection
    definition of, Definition
    distance from, Subsection
    eigenvectors of, Example
    existence of, Theorem
    is the closest vector, Subsection
    linearity of, Proposition
    of a vector in \(W\), Example
    of a vector in \(W^\perp\), Example
    onto a column space, Paragraphs
    onto a line, Subsection Subsection
    onto the \(xy\)-plane, Example Example Example
    noninvertibility of, Example
    properties of, Proposition
    range of, Proposition
    standard matrix of, Paragraph
    column space of, Proposition
    complicated matrix formula, Subsection
    diagonalizability of, Proposition
    eigenvalues of, Proposition
    eigenvectors of, Proposition
    noninvertibility of, Proposition
    null space of, Proposition
    properties of, Proposition
    square of, Proposition
    Orthogonal set
    and \(\cB\)-coordinates, Subsection
    and least squares, Subsection
    definition of, Definition
    linear independence of, Fact
    making orthonormal, Subsection
    necessity of, Subsection
    orthonormality of, Definition
    standard coordinate vectors, Subsection
    producing from a basis, Theorem
    Orthogonality
    and the Pythagorean theorem, Subsection
    definition of, Definition
    zero vector, Subsection
    Orthonormal set. See Orthogonal set
    Parallelepiped
    definition of, Definition
    flat, Note
    parallelogram, Subsection
    area of, Subsection
    volume of, Theorem
    Parallelogram. See Parallelepiped
    Parameterized equation, Note
    Parametric form, Subsection Subsection
    Parametric vector form
    of a homogeneous equation, Subsection Subsection
    of an inhomogeneous equation, Paragraph
    particular solution, Subsection
    Particular solution. See Parametric vector form
    Perron–Frobenius theorem, Theorem
    Pivot, Definition
    Pivot position, Definition
    Plane
    \(xy\)-plane, Subsection
    dimension-2 solution set, Subsection Subsection
    geometric definition of, Definition
    parametric form of, Subsection
    Point, Definition
    distance between, Definition
    Polynomial
    characteristic see Characteristic polynomial
    complex roots, Theorem
    conjugate roots, Appendix
    cubic, Appendix
    factoring by hand, Paragraphs
    quadratic, Appendix
    rational roots, Fact
    Power of a matrix. See Matrix multiplication
    Projection. See Orthogonal projection
    diagonalizability of, Example
    Projection Formula, Theorem
    Projection matrix. See Orthogonal projection, standard matrix of
    QR codes, Example
    Quadratic formula, Appendix
    Range. See Transformation
    Rank, Definition
    rank theorem, Theorem
    row and column, Subsection
    Rational Root Theorem, Fact
    Real \(n\)-space, Definition
    as a subspace of itself, Subsection
    point of, Definition
    Real numbers \(\R\), Paragraph Subsection
    Real part. See Complex numbers
    Red Box, Subsection Subsection Subsection
    Reduced row echelon form, Definition
    and invertibility, Theorem
    Reflection
    eigenspaces of, Example
    eigenvectors of, Subsection
    in general, Remark
    over the \(y\)-axis, Example Example
    inverse of, Example Example
    Rotation
    composition of, Example
    counterclockwise by \(45^\circ\), Example Example
    counterclockwise by \(90^\circ\), Example Example
    counterclockwise by \(\theta\), Example
    eigenvectors of, Example
    non-diagonalizability of, Example
    Rotation-scaling matrix
    and complex eigenvalues, Theorem
    computing the angle, Subsection
    definition of, Definition
    structure of, Proposition
    Rotation-Scaling Theorem, Theorem
    Row echelon form, Definition
    Row equivalence, Definition
    Row operations, Paragraphs
    and determinants, Subsection
    replacement, Item
    scaling, Item
    swap, Item
    Row rank. See Rank
    Row reduction
    algorithm, Algorithm
    computing determinants, Subsection
    picture of, Paragraph
    Row replacement. See Row operations, replacement
    Row space
    definition of, Definition
    is column space of transpose, Paragraph
    orthogonal complement of, Subsection
    Row vector. See Vector
    Scale. See Dilation; Row operations, scaling
    Set builder notation, Note
    Shear
    eigenvectors of, Example
    in the \(x\)-direction, Example Example
    multiplicities, Example
    non-diagonalizability of, Example
    Similarity
    action on a vector, Subsection
    and eigenspaces, Subsection
    and eigenvalues, Subsection
    and eigenvectors, Theorem
    and multiplicities, Theorem
    and powers, Fact Fact
    and the characteristic polynomial, Theorem
    and the determinant, Paragraph
    and the trace, Paragraph
    definition of, Definition Definition
    equivalence relation, Proposition
    geometry of, Subsection
    identity matrix, Paragraph
    of \(2\times 2\) matrices, Remark
    of diagonal matrices, Example
    of diagonalizable matrices, Paragraphs
    to a diagonal matrix see Diagonalizability
    worked example, Subsection
    Solution. See System of linear equations
    Solution set
    definition of, Definition
    of a homogeneous system is a null space, Subsection
    of a homogeneous system is a span, Subsection
    picture of, Subsection
    size of, Subsection
    translate of a span, Subsection
    versus the column space, Subsection
    Space
    \(\mathbb{R}^3 \), Subsection
    color space, Example
    dimension-3 solution set, Subsection Subsection
    Span
    basis of see Basis
    definition of, Subsection
    is a subspace, Theorem
    orthogonal complement of, Paragraph Subsection
    pictures of, Paragraphs
    Standard coordinate vectors
    and matrix columns, Fact
    are unit vectors, Paragraph
    columns of the identity matrix, Definition
    definition of, Subsection
    orthonormality of, Subsection
    picture of, Paragraph
    Standard matrix. See Linear transformation
    Steady state. See Stochastic matrix
    Stochastic matrix
    definition of, Definition
    eigenvalues of, Fact
    long-term behavior of, Theorem
    picture of, Figure Figure
    steady state of, Definition
    computing, Subsection Subsection
    sum of entries of vector, Subsection
    Subset
    definition of, Definition
    set builder notation, Note
    versus subspace, Subsection
    Subspace
    and spans, Theorem
    definition of, Definition
    is a span, Theorem
    orthogonal complement of see Orthogonal complement
    real \(n\)-space, Subsection
    versus subset, Subsection
    writing as column or null space, Paragraphs
    zero, Subsection
    Superposition principle, Paragraph
    System of linear equations
    consistent, Definition
    picture of, Note
    span criterion, Note Subsection
    definition of, Paragraph
    four ways of writing, Note
    homogeneous, Definition
    trivial solution, Subsection
    inconsistent, Definition
    picture of, Figure
    RREF criterion, Theorem
    inhomogeneous, Definition
    nontrivial solution, Subsection
    and free variables, Observation
    and invertibility, Theorem
    finding, Subsection
    number of solutions of, Subsection
    parametric form of see Parametric form
    parametric vector form of see Parametric vector form
    particular solution of see Parametric vector form
    solution of, Definition
    solving with the inverse matrix, Theorem
    trivial solution, Subsection
    and invertibility, Theorem
    Tall matrix. See Matrix transformation
    Trace. See Matrix
    Transformation
    addition of, Definition
    as a machine, Paragraph
    associated to a matrix see Matrix transformation
    codomain of, Definition
    composition of, Definition
    noncommutativity of, Subsection
    order of operations, Example
    definition of, Definition
    domain of, Definition
    identity see Identity transformation
    invertible see Invertible transformation
    linear see Linear transformation
    of one variable, Example
    of several variables, Example
    one-to-one see One-to-one
    onto see Onto
    range of, Definition
    scalar multiplication of, Definition
    Transpose. See Matrix
    Triangle
    area of see Volume
    Trivial solution. See System of linear equations
    Unit cube, Subsection
    Unit vector
    and orthonormality, Definition
    definition of, Definition
    in the direction of a vector, Fact
    standard coordinate vectors, Paragraph
    Upper-triangular
    see Matrix, Definition
    Vector
    addition, Item
    parallelogram law, Paragraphs
    angle between, Remark
    definition of, Definition
    distance between, Definition
    length of, Fact
    linear combination of see Linear combination
    orthogonal see Orthogonality
    product with matrix see Matrix-vector product
    real and imaginary parts of, Paragraph
    row vector, Definition
    product with column vector, Definition
    scalar multiplication, Item
    picture of, Paragraphs
    subtraction
    picture of, Paragraphs
    unit vector see Unit vector
    unit vector in the direction of, Fact
    Vector equation
    consistent see System of linear equations, consistent
    definition of, Definition
    equivalence with matrix equation, Note
    equivalence with system of equations, Paragraph
    inconsistent see System of linear equations, inconsistent
    solving, Subsection
    Volume
    and length, Subsection
    of a parallelepiped, Theorem
    of a region, Theorem
    of a triangle, Example
    of an ellipse, Example
    signed, Remark
    Wide matrix. See Linear independence; Matrix transformation