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Discover Linear Algebra
Jeremy Sylvestre
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Front Matter
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Colophon
Author Biography
Preface
I
Systems of Equations and Matrices
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1
Systems of linear equations
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1.1
Discovery guide
1.2
Terminology and notation
1.3
Concepts
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1.3.1
System solutions
1.3.2
Determining solutions
1.4
Examples
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1.4.1
Row operations versus equation manipulations
1.5
Exercises
1.6
Reflect on your understanding
2
Solving systems using matrices
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2.1
Discovery guide
2.2
Terminology and notation
2.3
Concepts
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2.3.1
Reducing matrices
2.3.2
Solving systems
2.4
Examples
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2.4.1
Worked examples from the discovery guide
2.5
Theory
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2.5.1
Reduced matrices
2.5.2
Solving systems using matrices
2.6
Exercises
2.7
Reflect on your understanding
3
Using systems of equations
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3.1
Discovery guide
3.2
Examples
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3.2.1
A simple example
3.2.2
Flow in networks
3.2.3
Balancing chemical equations
3.2.4
Polynomial interpolation
3.3
Terminology and notation
3.4
Theory
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3.4.1
Polynomial interpolation
4
Matrices and matrix operations
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4.1
Discovery guide
4.2
Terminology and notation
4.3
Concepts
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4.3.1
Matrix entries
4.3.2
Matrix dimensions
4.3.3
Matrix equality
4.3.4
Basic matrix operations
4.3.5
The zero matrix
4.3.6
Linear systems as matrix equations
4.3.7
Matrix multiplication
4.3.8
Matrix powers
4.3.9
Transpose
4.4
Examples
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4.4.1
Basic matrix operations
4.4.2
Matrix multiplication
4.4.3
Combining operations
4.4.4
Linear systems as matrix equations
4.4.4.1
A first example
4.4.4.2
Expressing system solutions in vector form
4.4.5
Transpose
4.5
Theory
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4.5.1
Rules of matrix algebra
4.5.2
Linear systems as matrix equations
4.6
Exercises
4.7
Reflect on your understanding
5
Matrix inverses
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5.1
Discovery guide
5.2
Terminology and notation
5.3
Concepts
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5.3.1
The identity matrix
5.3.2
Inverse matrices
5.3.3
Matrix division
5.3.4
Cancellation
5.3.5
Solving systems using inverses
5.4
Examples
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5.4.1
Inverses of
2
×
2
matrices
5.4.2
Solving systems using inverses
5.4.3
Solving other matrix equations using inverses
5.5
Theory
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5.5.1
Properties of the identity matrix
5.5.2
Properties of the inverse
5.6
Exercises
5.7
Reflect on your understanding
6
Elementary matrices
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6.1
Discovery guide
6.2
Terminology and notation
6.3
Concepts
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6.3.1
Elementary matrices
6.3.2
Inverses by elementary matrices
6.3.3
Inverses of elementary matrices
6.3.4
Decomposition of invertible matrices
6.3.5
Inverses by row reduction
6.4
Examples
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6.4.1
Elementary matrices and their inverses
6.4.2
Decomposing an invertible matrix and its inverse into elementary matrices
6.4.3
Inversion by row reduction
6.5
Theory
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6.5.1
Inverses of elementary matrices
6.5.2
Inverses versus row operations
6.5.3
More properties of inverses
6.5.4
Solution sets of row equivalent matrices
6.6
Exercises
6.7
Reflect on your understanding
7
Special forms of square matrices
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7.1
Discovery guide
7.2
Terminology and notation
7.3
Concepts
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7.3.1
Algebra with scalar matrices
7.3.2
Inverses of special forms
7.3.3
Decompositions using special forms
7.4
Examples
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7.4.1
Computation patterns
7.5
Theory
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7.5.1
Algebra of special forms
7.5.2
Invertibility of special forms
7.6
Exercises
7.7
Reflect on your understanding
8
Determinants
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8.1
Discovery guide
8.2
Terminology and notation
8.3
Concepts
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8.3.1
Definition of the determinant
8.3.2
Determinants of
1
×
1
matrices
8.3.3
Determinants of
2
×
2
matrices
8.3.4
Determinants of larger matrices
8.3.5
Determinants of special forms
8.4
Examples
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8.4.1
Determinants of
2
×
2
matrices
8.4.2
Minors and cofactors of
3
×
3
matrices
8.4.2.1
Minors
8.4.2.2
Cofactors
8.4.3
Determinants of
3
×
3
matrices
8.4.4
Minors and cofactors of
4
×
4
matrices
8.4.5
Determinants of
4
×
4
matrices
8.5
Theory
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8.5.1
Basic properties of determinants
8.6
Exercises
8.7
Reflect on your understanding
9
Determinants versus row operations
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9.1
Discovery guide
9.2
Concepts
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9.2.1
Effect of row operations on the determinant
9.2.1.1
Swapping rows
9.2.1.2
Multiplying rows
9.2.1.3
Combining rows
9.2.2
Column operations and the transpose
9.2.3
Determinants by row reduction
9.3
Examples
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9.3.1
Determinants by row reduction
9.3.2
Matrices of determinant zero
9.4
Theory
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9.4.1
Effect of row operations on the determinant
9.4.2
Determinants of elementary matrices
9.5
Exercises
9.6
Reflect on your understanding
10
Determinants, the adjoint, and inverses
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10.1
Discovery guide
10.2
Terminology and notation
10.3
Concepts
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10.3.1
The classical adjoint
10.3.2
Determinants determine invertibility
10.3.3
Determinants versus matrix multiplication
10.3.3.1
Case of elementary matrices
10.3.3.2
Invertible case
10.3.3.3
Singular case
10.3.3.4
All cases
10.3.4
Determinant of an inverse
10.3.5
Cramer’s rule
10.4
Examples
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10.4.1
The
2
×
2
case
10.4.2
Computing an inverse using the adjoint
10.4.3
Cramer’s rule
10.5
Theory
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10.5.1
Adjoints and inverses
10.5.2
Determinants determine invertibility
10.5.3
Determinant formulas
10.5.4
Cramer’s rule
10.6
Exercises
10.7
Reflect on your understanding
11
Complex systems and matrices
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11.1
Motivation
11.2
Terminology and notation
11.3
Examples
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11.3.1
Complex linear systems
11.3.2
Complex matrices
11.3.2.1
Inverses
11.3.2.2
Determinant
11.3.2.3
Conjugate and the complex adjoint
11.4
Theory
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11.4.1
Complex linear systems
11.4.2
Complex matrices
11.4.2.1
Basic algebra
11.4.2.2
Inverses
11.4.2.3
Determinants
11.4.2.4
Self-adjoint matrices
II
Vector Spaces
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12
Introduction to vectors
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12.1
Discovery guide
12.2
Terminology and notation
12.3
Concepts
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12.3.1
Vectors
12.3.2
Vector addition
12.3.3
The zero vector
12.3.4
Vector negatives and vector subtraction
12.3.5
Scalar multiplication
12.3.6
Vector algebra
12.3.7
The standard basis vectors
12.4
Examples
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12.4.1
Vectors in
R
n
12.4.2
Vector operations
12.5
Theory
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12.5.1
Vector algebra
13
Geometry of vectors
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13.1
Discovery guide
13.2
Terminology and notation
13.3
Concepts
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13.3.1
Geometric length of a vector: the norm
13.3.2
Properties of the norm
13.3.3
Unit vectors and normalization
13.3.4
Distance between vectors
13.3.5
Angle between vectors in the plane and in space
13.3.6
Dot product
13.3.7
Angle between vectors in
R
n
13.3.8
Dot product versus norm
13.3.9
Dot product as matrix multiplication
13.4
Examples
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13.4.1
The norm of a vector
13.4.2
Dot product and the angle between vectors
13.5
Theory
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13.5.1
Norm and dot product
13.5.2
Vector geometry inequalities and uniqueness of vector angles
13.5.2.1
The Cauchy-Schwarz inequality
13.5.2.2
The triangle inequality
14
Orthogonal vectors
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14.1
Discovery guide
14.2
Terminology and notation
14.3
Concepts
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14.3.1
Values of
u
⋅
v
14.3.2
Orthogonal vectors
14.3.2.1
Orthogonal vectors in
R
2
14.3.3
Orthogonal projection
14.3.4
Normal vectors of lines in the plane
14.3.5
Normal vectors of planes in space
14.3.6
The cross product
14.4
Examples
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14.4.1
Orthogonal vectors
14.4.2
Orthogonal projection
14.4.3
Cross product
14.5
Theory
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14.5.1
Properties of orthogonal vectors and orthogonal projection
14.5.2
Decomposition of a vector into orthogonal components
14.5.3
Properties of the cross product
15
Geometry of linear systems
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15.1
Discovery guide
15.2
Terminology and notation
15.3
Concepts
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15.3.1
Lines in the plane
15.3.2
Lines in space
15.3.3
Planes in space
15.3.4
Parallel vectors as a “basis” for lines and planes
15.3.5
Summary
15.4
Examples
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15.4.1
Describing lines and planes parametrically
15.4.2
Determining points of intersection
16
Abstract vector spaces
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16.1
Discovery guide
16.2
Motivation
16.3
Terminology and notation
16.4
Concepts
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16.4.1
The ten vector space axioms
16.4.2
Instances of vector spaces
16.5
Examples
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16.5.1
Verifying axioms: the space of positive numbers
16.5.2
Verifying axioms: the space of functions
16.6
Theory
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16.6.1
Uniqueness of the zero vector and of negatives
16.6.2
Basic vector algebra rules
17
Subspaces
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17.1
Discovery guide
17.2
Terminology and notation
17.3
Concepts
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17.3.1
Recognizing subspaces
17.3.2
Building subspaces
17.3.3
The subspaces of
R
n
17.3.4
Recognizing when two subspaces are the same
17.4
Examples
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17.4.1
The Subspace Test
17.4.2
Important subspace examples
17.4.3
Determining if a vector is in a span
17.4.4
Determining if a spanning set generates the whole vector space
17.5
Theory
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17.5.1
The Subspace Test
17.5.2
Universal examples of subspaces
17.5.3
Equality of subspaces created via spanning sets
17.6
More examples
18
Linear independence
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18.1
Discovery guide
18.2
Terminology and notation
18.3
Concepts
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18.3.1
Reducing spanning sets
18.3.2
Linear dependence and independence
18.3.3
Linear dependence and independence of just one or two vectors
18.3.4
Linear dependence and independence in
R
n
18.4
Examples
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18.4.1
Testing dependence/independence
18.4.2
Linear independence of “standard” spanning sets
18.5
Theory
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18.5.1
Basic facts about linear dependence and independence
18.5.2
Linear dependence and independence of spanning sets
19
Basis and Coordinates
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19.1
Discovery guide
19.2
Terminology and notation
19.3
Concepts
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19.3.1
Basis as a minimal spanning set
19.3.2
Basis as a maximal linearly independent set
19.3.3
Basis is not unique
19.3.4
Ordered versus unordered basis
19.3.5
Coordinates of a vector
19.3.5.1
Basic concept of coordinates relative to a basis
19.3.5.2
Linearity of coordinates
19.4
Examples
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19.4.1
Checking a basis
19.4.2
Standard bases
19.4.3
Coordinate vectors
19.5
Theory
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19.5.1
Reducing to a basis
19.5.2
Basis as optimal spanning set
20
Dimension
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20.1
Discovery guide
20.2
Terminology and notation
20.3
Concepts
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20.3.1
The “just-right” number of vectors in a spanning set
20.3.2
Dimension as geometric “degrees of freedom”
20.3.3
Dimension as algebraic “degrees of freedom”
20.3.4
The dimension of a subspace
20.3.5
The dimension of the trivial vector space
20.4
Examples
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20.4.1
Determining a basis from a parametric expression
20.4.2
An infinite-dimensional example
20.4.3
Enlarging a linearly independent set to a basis
20.5
Theory
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20.5.1
Dimension as size of a basis
20.5.2
Consequences for the theory of linear dependence/independence and spanning
20.5.3
Dimension of subspaces
21
Column, row, and null spaces
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21.1
Discovery guide
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21.1.1
Column space
21.1.2
Row space
21.1.3
Null space
21.1.4
Relationship between the three spaces
21.2
Terminology and notation
21.3
Concepts
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21.3.1
Column space
21.3.2
Row space
21.3.3
Column space versus row space
21.3.4
Null space and the dimensions of the three spaces
21.4
Examples
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21.4.1
The three spaces
21.4.2
Enlarging a linearly independent set
21.5
Theory
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21.5.1
Column space
21.5.2
Row space
21.5.3
Column and row spaces versus rank and invertibility
22
Change of basis
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22.1
Discovery guide
22.2
Terminology and notation
22.3
Concepts
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22.3.1
Linearity of coordinate vectors
22.3.2
Matrix-times-vector
as a linear combination
22.3.3
Converting coordinate vectors
22.3.4
Properities of transition matrices
22.3.5
Change of basis in
R
n
22.3.5.1
Changing to the standard basis
22.3.5.2
Computing transition matrices using the standard basis as an intermediate
22.3.5.3
Computing transition matrices using row reduction
22.4
Examples
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22.4.1
Computing a transition matrix
22.4.2
Computing a transition matrix for
R
n
22.5
Theory
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22.5.1
Linearity properties of coordinate vectors
22.5.2
Properties of transition matrices
22.5.3
Change of basis in
R
n
23
Complex vector spaces
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23.1
Motivation
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23.1.1
Complex
n
-dimensional vectors
23.1.2
Complex abstract vectors
23.2
Terminology and notation
23.3
Concepts
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23.3.1
The vector space
C
n
23.3.2
Complex vector spaces in general
23.3.3
Every complex space is also a real space
23.4
Examples
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23.4.1
Instances of complex vector spaces
23.4.2
Calculations in complex vector spaces
III
Introduction to Matrix Forms
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24
Eigenvalues and eigenvectors
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24.1
Discovery guide
24.2
Terminology and notation
24.3
Motivation
24.4
Concepts
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24.4.1
Determining eigenvalues
24.4.2
Eigenvalues for special forms of matrices
24.4.3
Determining eigenvectors
24.4.4
Eigenspaces
24.4.5
Connection to invertibility
24.4.6
The geometry of eigenvectors
24.5
Examples
24.6
Theory
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24.6.1
Basic facts
24.6.2
Eigenvalues and invertibility
25
Diagonalization
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25.1
Discovery guide
25.2
Terminology and notation
25.3
Motivation
25.4
Concepts
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25.4.1
The transition matrix and the diagonal form
25.4.2
Diagonalizable matrices
25.4.3
Diagonalization procedure
25.5
Examples
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25.5.1
Carrying out the diagonalization procedure
25.5.2
Determining diagonalizability from multiplicities
25.5.3
A different kind of example
25.6
Theory
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25.6.1
Similar matrices
25.6.2
Diagonalizable matrices
25.6.3
The geometry of eigenvectors
25.6.4
More about diagonalizable matrices
26
Similarity
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26.1
Discovery guide
26.2
Terminology and notation
26.3
Concepts
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26.3.1
The geometry of similarity
26.3.2
The algebra of similarity
26.3.3
Similarity classes
26.3.4
The similarities of similar matrices
26.4
Examples
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26.4.1
The algebraic pattern of similarity
26.4.2
Computing
P
−
1
A
P
by row reduction
26.5
Theory
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26.5.1
Similarity is an equivalence relation
26.5.2
Properties of similar matrices
27
Application to systems of differential equations
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27.1
Discovery guide
27.2
Terminology and notation
27.3
Concepts
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27.3.1
Change of variables in a system of first-order, linear differential equations
27.3.2
Systems with a diagonalizable coefficient matrix
27.3.3
Homogeneous, second-order, linear differential equations
27.3.4
Systems with complex eigenvalues
27.4
Examples
28
Block-diagonal form
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28.1
Motivation
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28.1.1
Looking back to look forward
28.1.2
Generalizing the diagonalizable case
28.2
Discovery guide
28.3
Terminology and notation
28.4
Concepts
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28.4.1
Operations with block matrices
28.4.2
Properties of block-diagonal matrices
28.4.3
Invariant subspaces
28.4.4
Independent subspaces
28.4.4.1
Basic concept
28.4.4.2
Independent subspaces in
R
3
28.4.5
The similarity pattern of block-diagonal form
28.4.6
Block-diagonalization procedure
28.5
Examples
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28.5.1
Operations with block matrices
28.5.2
Putting a matrix in block-diagonal form
28.6
Theory
chevron_left
28.6.1
Properties of block-diagonal form
28.6.2
Invariant subspaces
28.6.3
Independent subspaces
29
Scalar-triangular form
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29.1
Motivation
29.2
Discovery guide
29.3
Terminology and notation
29.4
Concepts
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29.4.1
Scalar-triangular form
29.4.2
Generalized eigenvectors
29.4.3
The similarity pattern of scalar-triangular form
29.4.4
Scalar-triangularization procedure
29.5
Examples
29.6
Theory
30
Triangular block form
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30.1
Motivation
30.2
Examples
30.3
Terminology and notation
30.4
Concepts
30.5
Theory
chevron_left
30.5.1
Similarity to triangular block form
30.5.2
Properties of generalized eigenvectors
31
Consequences of triangular block form
chevron_left
31.1
Discovery guide
31.2
Terminology and notation
31.3
Concepts
chevron_left
31.3.1
Matrix polynomials
31.3.2
Nilpotent matrices
31.3.3
Matrices and their characteristic polynomials
31.3.4
Connection to determinants and traces of matrices
31.4
Examples
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31.4.1
Matrix polynomials
31.4.2
Nilpotent matrices
31.5
Theory
chevron_left
31.5.1
Nilpotent matrices
31.5.2
The Cayley-Hamilton Theorem
31.5.3
More consequences of triangular block form
32
Elementary nilpotent form
chevron_left
32.1
Discovery guide
32.2
Motivation
32.3
Terminology and notation
32.4
Concepts
chevron_left
32.4.1
Moving past triangular block form
32.4.2
The similarity pattern of elementary nilpotent form
32.4.3
Elementary nilpotent procedure
32.4.4
Cyclic subspaces
32.4.5
Properties of elementary nilpotent matrices
32.5
Examples
32.6
Theory
33
Triangular-block nilpotent form
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33.1
Motivation
33.2
Terminology and notation
33.3
Discovery guide
33.4
Concepts
chevron_left
33.4.1
From elementary to block nilpotent form
33.4.2
Determining the form indirectly
33.4.3
Procedures for any nilpotent matrix
33.5
Examples
chevron_left
33.5.1
Determining the form indirectly
33.5.2
Determining a transition matrix
33.6
Theory
34
Jordan normal form
chevron_left
34.1
Motivation
34.2
Terminology and notation
34.3
Concepts
chevron_left
34.3.1
Properties of Jordan normal form
34.3.2
Uniqueness of Jordan normal form
34.3.3
Jordan normal form procedure
34.3.4
Determining the form indirectly
34.4
Examples
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34.4.1
Determining the form indirectly
34.4.2
Using the procedure
34.5
Theory
35
Summary of matrix forms
chevron_left
35.1
Diagonal form
35.2
Block-diagonal form
35.3
Scalar-triangular form
35.4
Triangular-block form
35.5
Elementary nilpotent form
35.6
Triangular-block nilpotent form
35.7
Jordan normal form
IV
Inner Product Spaces
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36
Abstract inner product spaces
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36.1
Motivation
36.2
Discovery guide
36.3
Terminology and notation
36.4
Concepts
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36.4.1
Inner product axioms
36.4.2
Geometry in real inner product spaces
36.4.3
Norm and the dot product in
C
n
36.4.4
Complex inner products
36.4.5
Geometry in complex inner product spaces
36.4.6
Dot products as matrix multiplication
36.4.7
Other inner products on
R
n
and
C
n
36.5
Examples
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36.5.1
Inner products on familiar spaces
36.5.2
Geometry in inner product spaces
36.5.3
Skewing geometry in
R
n
36.6
Theory
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36.6.1
Properties of inner products
36.6.2
Inner products of
R
n
and
C
n
37
Orthgonality
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37.1
Discovery guide
37.2
Terminology and notation
37.3
Concepts
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37.3.1
Orthogonal complements
37.3.2
Expansion relative to an orthogonal basis
37.3.3
The Gram-Schmidt orthogonalization process
37.3.4
Orthogonal complements from an orthogonal basis
37.4
Examples
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37.4.1
Orthogonal complements
37.4.2
Expansion relative to an orthogonal basis
37.4.3
Using the Gram-Schmidt orthogonalization process
37.4.4
Obtaining an orthogonal complement using the Gram-Schmidt process
37.4.5
An infinite orthogonal set
37.5
Theory
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37.5.1
Orthogonal sets
37.5.2
Orthogonal bases
37.5.3
Orthogonal complements
38
Orthogonal projection and best approximation
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38.1
Concepts
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38.1.1
Orthogonal projection
38.1.2
Gram-Schmidt process versus orthogonal projection
38.1.3
Best approximation and distance between vector and subspace
38.1.4
Least squares solutions to a linear system
38.2
Terminology and notation
38.3
Examples
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38.3.1
Orthogonal projection
38.3.2
Best approximation
38.3.3
Least-squares solutions to an inconsistent system
38.4
Theory
chevron_left
38.4.1
Properties of orthogonal projection
38.4.2
Best approximation is best
38.4.3
Normal system is consistent
39
Matrix adjoints
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39.1
Discovery guide
39.2
Terminology and notation
39.3
Concepts
chevron_left
39.3.1
Adjoint matrices
39.3.2
Self-adjoint matrices
39.3.3
Orthogonal and unitary matrices
39.4
Examples
39.5
Theory
chevron_left
39.5.1
Properties of adjoints
39.5.2
Properties of product-preserving matrices
40
Orthogonal/unitary diagonalization
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40.1
Motivation
40.2
Discovery guide
40.3
Terminology and notation
40.4
Concepts
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40.4.1
Diagonalization of Hermitian and symmetric matrices
40.4.2
Diagonalization of normal matrices
40.5
Examples
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40.5.1
Orthogonal diagonalization of a symmetric matrix
40.5.2
Unitary diagonalization of a Hermitian matrix
40.5.3
Unitary diagonalization of a normal matrix
40.6
Theory
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40.6.1
Eigenvalues and eigenvectors
40.6.2
Characterizations of orthogonal/unitary diagonalization
40.6.3
Special instances of normal matrices
41
Quadratic forms
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41.1
Discovery guide
41.2
Terminology and notation
41.3
Concepts
chevron_left
41.3.1
Quadratic forms
41.3.2
Diagonalizing quadratic forms
41.3.3
Principal axes for a quadratic form
41.4
Examples
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41.4.1
Quadratic forms represented by matrices
41.4.2
Level curves of two-variable quadratic forms
41.5
Theory
V
Linear Transformations
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42
Matrix and linear transformations
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42.1
Discovery guide
42.2
Terminology and notation
42.3
Concepts
chevron_left
42.3.1
Matrix transformations
42.3.2
Linear transformations
42.3.3
Spanning set and basis image vectors
42.3.4
The standard matrix of a transformation
R
n
→
R
m
42.3.5
Important examples
42.3.6
The space of transformations
V
→
W
42.3.7
Linear functionals
42.3.8
Linear transformations of complex vector spaces
42.4
Examples
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42.4.1
Verifying the axioms
42.4.2
The standard matrix of a transformation
R
n
→
R
m
42.4.3
Linear transformations via spanning image vectors
42.5
Theory
chevron_left
42.5.1
Properties of linear transformations
42.5.2
Spaces of linear transformations
43
Kernel and image
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43.1
Discovery guide
43.2
Terminology and notation
43.3
Concepts
chevron_left
43.3.1
Kernel and image of matrix transformations
43.3.2
Bases for kernel and image
43.3.3
Dimensions of the kernel and image of a transformation
43.4
Examples
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43.4.1
Kernel and image of a matrix transformation
43.4.2
Kernel and image of a linear transformation
43.4.3
Special examples
43.5
Theory
chevron_left
43.5.1
Kernel and image are subspaces
43.5.2
Basis and dimension of kernel and image
44
Compositions and isomorphisms
chevron_left
44.1
Discovery guide
44.2
Terminology and notation
44.3
Concepts
chevron_left
44.3.1
Composition of linear transformations
44.3.2
Composition of matrix transformations
44.3.3
Inverse transformations
44.3.4
Invertibility conditions
44.3.5
Inverses of matrix transformations
44.3.6
Constructing invertible transformations
44.3.7
Isomorphisms
44.3.8
Constructing isomorphisms
44.3.9
Important isomorphisms
44.4
Examples
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44.4.1
Computing with compositions
44.4.2
Checking and computing inverses
44.4.3
Checking surjectivity
44.4.4
Defining isomorphisms by choice of bases
44.5
Theory
chevron_left
44.5.1
Properties of composite transformations
44.5.2
One-to-one and invertible transformations
44.5.3
Isomorphisms
45
The matrix of a linear transformation
chevron_left
45.1
Discovery guide
45.2
Terminology and notation
45.3
Concepts
chevron_left
45.3.1
Attaching a matrix to a transformation
45.3.2
Computing the matrix of a transformation
45.3.3
Important examples
45.3.4
Matrices of compositions and inverses
45.3.5
Properties of a transformation reflected in its matrix
45.4
Examples
chevron_left
45.4.1
Computing the matrix of a linear transformation
45.4.2
Using the matrix of a linear transformation
45.5
Theory
chevron_left
45.5.1
The matrix of a linear transformation
45.5.2
Properties of a transformation from properties of its matrix
45.5.3
Matrices of compositions and inverses
45.5.4
The space of linear transformations as a space of matrices
46
Similarity of linear operators
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46.1
Terminology and notation
46.2
Concepts
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46.2.1
Matrices of an operator
46.2.2
Transferring matrix concepts to operators
46.2.3
Canonical forms of linear operators
46.3
Examples
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46.3.1
Using a “natural” basis for an operator
46.3.2
Computing determinant, eigenvalues, and eigenvectors of operators
46.4
Theory
Back Matter
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A
Complex numbers
chevron_left
A.1
Motivation
A.2
The field of complex numbers
chevron_left
A.2.1
Definition of the field
A.2.2
Basic operations with complex numbers
A.2.3
Division of complex nubers
A.2.4
Properties of complex numbers
A.2.5
Complex polynomials
A.3
The complex plane
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A.3.1
Visualizing the complex numbers
A.3.2
Polar coordinates
A.3.3
Roots of unity
B
Sage Tutorials
chevron_left
B.1
Basics
B.2
Linear algebra basics
B.3
Diagonal form
B.4
Scalar-triangular form
chevron_left
B.4.1
A
3
×
3
example
B.4.2
A
5
×
5
example
B.5
Triangular block form
chevron_left
B.5.1
Matrix and eigenvalues
B.5.2
Analysis of
λ
1
B.5.3
Analysis of
λ
2
B.5.4
The transition matrix and the triangular block form matrix
B.6
Gram-Schmidt orthogonalization
chevron_left
B.6.1
An example in
R
4
B.6.2
An example in
C
4
B.6.3
A polynomial example with an integral inner product
B.6.4
A polynomial example with a “sampling” inner product
B.6.5
Bonus Fun
B.7
Best approximation
chevron_left
B.7.1
Approximating a matrix
B.7.2
Approximating a function
B.8
Orthogonal/unitary diagonalization
chevron_left
B.8.1
Orthogonally diagonalizing a symmetric matrix
B.8.2
Unitarily diagonalizing a normal matrix
C
GNU Free Documentation License
Bibliography
Index
Colophon
Chapter
26
Similarity
26.1
Discovery guide
26.2
Terminology and notation
26.3
Concepts
26.4
Examples
26.5
Theory
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