7.5: Index
- Page ID
- 70221
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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 1\)">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
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
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 transformation
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
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
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
Rotation
composition of, 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
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 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
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