Skip to main content
Discover Linear Algebra:
A first course in linear algebra
Jeremy Sylvestre
Contents
Index
Search Book
close
Search Results:
No results.
dark_mode
Dark Mode
Prev
Up
Next
Front Matter
chevron_left
Colophon
Author Biography
Preface
I
Systems of Equations and Matrices
chevron_left
1
Systems of linear equations
chevron_left
1.1
Discovery guide
1.2
Terminology and notation
1.3
Concepts
chevron_left
1.3.1
System solutions
1.3.2
Determining solutions
1.4
Examples
chevron_left
1.4.1
Row operations versus equation manipulations
1.5
Exercises
1.6
Reflect on your understanding
2
Solving systems using matrices
chevron_left
2.1
Discovery guide
2.2
Terminology and notation
2.3
Concepts
chevron_left
2.3.1
Reducing matrices
2.3.2
Solving systems
2.4
Examples
chevron_left
2.4.1
Worked examples from the discovery guide
2.5
Theory
chevron_left
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
chevron_left
3.1
Discovery guide
3.2
Examples
chevron_left
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
chevron_left
3.4.1
Polynomial interpolation
4
Matrices and matrix operations
chevron_left
4.1
Discovery guide
4.2
Terminology and notation
4.3
Concepts
chevron_left
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
chevron_left
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
chevron_left
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
chevron_left
5.1
Discovery guide
5.2
Terminology and notation
5.3
Concepts
chevron_left
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
chevron_left
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
chevron_left
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
chevron_left
6.1
Discovery guide
6.2
Terminology and notation
6.3
Concepts
chevron_left
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
chevron_left
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
chevron_left
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
chevron_left
7.1
Discovery guide
7.2
Terminology and notation
7.3
Concepts
chevron_left
7.3.1
Algebra with scalar matrices
7.3.2
Inverses of special forms
7.3.3
Decompositions using special forms
7.4
Examples
chevron_left
7.4.1
Computation patterns
7.5
Theory
chevron_left
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
chevron_left
8.1
Discovery guide
8.2
Terminology and notation
8.3
Concepts
chevron_left
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
chevron_left
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
chevron_left
8.5.1
Basic properties of determinants
8.6
Exercises
8.7
Reflect on your understanding
9
Determinants versus row operations
chevron_left
9.1
Discovery guide
9.2
Concepts
chevron_left
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
chevron_left
9.3.1
Determinants by row reduction
9.3.2
Matrices of determinant zero
9.4
Theory
chevron_left
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
chevron_left
10.1
Discovery guide
10.2
Terminology and notation
10.3
Concepts
chevron_left
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
chevron_left
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
chevron_left
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
II
Vector Spaces
chevron_left
11
Introduction to vectors
chevron_left
11.1
Discovery guide
11.2
Terminology and notation
11.3
Concepts
chevron_left
11.3.1
Vectors
11.3.2
Vector addition
11.3.3
The zero vector
11.3.4
Vector negatives and vector subtraction
11.3.5
Scalar multiplication
11.3.6
Vector algebra
11.3.7
The standard basis vectors
11.4
Examples
chevron_left
11.4.1
Vectors in
R
n
11.4.2
Vector operations
11.5
Theory
chevron_left
11.5.1
Vector algebra
12
Geometry of vectors
chevron_left
12.1
Discovery guide
12.2
Terminology and notation
12.3
Concepts
chevron_left
12.3.1
Geometric length of a vector: the norm
12.3.2
Properties of the norm
12.3.3
Unit vectors and normalization
12.3.4
Distance between vectors
12.3.5
Angle between vectors in the plane and in space
12.3.6
Dot product
12.3.7
Angle between vectors in
R
n
12.3.8
Dot product versus norm
12.3.9
Dot product as matrix multiplication
12.4
Examples
chevron_left
12.4.1
The norm of a vector
12.4.2
Dot product and the angle between vectors
12.5
Theory
chevron_left
12.5.1
Norm and dot product
12.5.2
Vector geometry inequalities and uniqueness of vector angles
12.5.2.1
The Cauchy-Schwarz inequality
12.5.2.2
The triangle inequality
13
Orthogonal vectors
chevron_left
13.1
Discovery guide
13.2
Terminology and notation
13.3
Concepts
chevron_left
13.3.1
Values of
u
⋅
v
13.3.2
Orthogonal vectors
13.3.2.1
Orthogonal vectors in
R
2
13.3.3
Orthogonal projection
13.3.4
Normal vectors of lines in the plane
13.3.5
Normal vectors of planes in space
13.3.6
The cross product
13.4
Examples
chevron_left
13.4.1
Orthogonal vectors
13.4.2
Orthogonal projection
13.4.3
Cross product
13.5
Theory
chevron_left
13.5.1
Properties of orthogonal vectors and orthogonal projection
13.5.2
Decomposition of a vector into orthogonal components
13.5.3
Properties of the cross product
14
Geometry of linear systems
chevron_left
14.1
Discovery guide
14.2
Terminology and notation
14.3
Concepts
chevron_left
14.3.1
Lines in the plane
14.3.2
Lines in space
14.3.3
Planes in space
14.3.4
Parallel vectors as a “basis” for lines and planes
14.3.5
Summary
14.4
Examples
chevron_left
14.4.1
Describing lines and planes parametrically
14.4.2
Determining points of intersection
15
Abstract vector spaces
chevron_left
15.1
Discovery guide
15.2
Motivation
15.3
Terminology and notation
15.4
Concepts
chevron_left
15.4.1
The ten vector space axioms
15.4.2
Instances of vector spaces
15.5
Examples
chevron_left
15.5.1
Verifying axioms: the space of positive numbers
15.5.2
Verifying axioms: the space of functions
15.6
Theory
chevron_left
15.6.1
Uniqueness of the zero vector and of negatives
15.6.2
Basic vector algebra rules
16
Subspaces
chevron_left
16.1
Discovery guide
16.2
Terminology and notation
16.3
Concepts
chevron_left
16.3.1
Recognizing subspaces
16.3.2
Building subspaces
16.3.3
The subspaces of
R
n
16.3.4
Recognizing when two subspaces are the same
16.4
Examples
chevron_left
16.4.1
The Subspace Test
16.4.2
Important subspace examples
16.4.3
Determining if a vector is in a span
16.4.4
Determining if a spanning set generates the whole vector space
16.5
Theory
chevron_left
16.5.1
The Subspace Test
16.5.2
Universal examples of subspaces
16.5.3
Equality of subspaces created via spanning sets
16.6
More examples
17
Linear independence
chevron_left
17.1
Discovery guide
17.2
Terminology and notation
17.3
Concepts
chevron_left
17.3.1
Reducing spanning sets
17.3.2
Linear dependence and independence
17.3.3
Linear dependence and independence of just one or two vectors
17.3.4
Linear dependence and independence in
R
n
17.4
Examples
chevron_left
17.4.1
Testing dependence/independence
17.4.2
Linear independence of “standard” spanning sets
17.5
Theory
chevron_left
17.5.1
Basic facts about linear dependence and independence
17.5.2
Linear dependence and independence of spanning sets
18
Basis and Coordinates
chevron_left
18.1
Discovery guide
18.2
Terminology and notation
18.3
Concepts
chevron_left
18.3.1
Basis as a minimal spanning set
18.3.2
Basis as a maximal linearly independent set
18.3.3
Basis is not unique
18.3.4
Ordered versus unordered basis
18.3.5
Coordinates of a vector
18.3.5.1
Basic concept of coordinates relative to a basis
18.3.5.2
Linearity of coordinates
18.4
Examples
chevron_left
18.4.1
Checking a basis
18.4.2
Standard bases
18.4.3
Coordinate vectors
18.5
Theory
chevron_left
18.5.1
Reducing to a basis
18.5.2
Basis as optimal spanning set
19
Dimension
chevron_left
19.1
Discovery guide
19.2
Terminology and notation
19.3
Concepts
chevron_left
19.3.1
The “just-right” number of vectors in a spanning set
19.3.2
Dimension as geometric “degrees of freedom”
19.3.3
Dimension as algebraic “degrees of freedom”
19.3.4
The dimension of a subspace
19.3.5
The dimension of the trivial vector space
19.4
Examples
chevron_left
19.4.1
Determining a basis from a parametric expression
19.4.2
An infinite-dimensional example
19.4.3
Enlarging a linearly independent set to a basis
19.5
Theory
chevron_left
19.5.1
Dimension as size of a basis
19.5.2
Consequences for the theory of linear dependence/independence and spanning
19.5.3
Dimension of subspaces
20
Column, row, and null spaces
chevron_left
20.1
Discovery guide
chevron_left
20.1.1
Column space
20.1.2
Row space
20.1.3
Null space
20.1.4
Relationship between the three spaces
20.2
Terminology and notation
20.3
Concepts
chevron_left
20.3.1
Column space
20.3.2
Row space
20.3.3
Column space versus row space
20.3.4
Null space and the dimensions of the three spaces
20.4
Examples
chevron_left
20.4.1
The three spaces
20.4.2
Enlarging a linearly independent set
20.5
Theory
chevron_left
20.5.1
Column space
20.5.2
Row space
20.5.3
Column and row spaces versus rank and invertibility
III
Introduction to Matrix Forms
chevron_left
21
Eigenvalues and eigenvectors
chevron_left
21.1
Discovery guide
21.2
Terminology and notation
21.3
Motivation
21.4
Concepts
chevron_left
21.4.1
Determining eigenvalues
21.4.2
Eigenvalues for special forms of matrices
21.4.3
Determining eigenvectors
21.4.4
Eigenspaces
21.4.5
Connection to invertibility
21.4.6
The geometry of eigenvectors
21.5
Examples
21.6
Theory
chevron_left
21.6.1
Basic facts
21.6.2
Eigenvalues and invertibility
22
Diagonalization
chevron_left
22.1
Discovery guide
22.2
Terminology and notation
22.3
Motivation
22.4
Concepts
chevron_left
22.4.1
The transition matrix and the diagonal form
22.4.2
Diagonalizable matrices
22.4.3
Diagonalization procedure
22.5
Examples
chevron_left
22.5.1
Carrying out the diagonalization procedure
22.5.2
Determining diagonalizability from multiplicities
22.5.3
A different kind of example
22.6
Theory
chevron_left
22.6.1
Similar matrices
22.6.2
Diagonalizable matrices
22.6.3
The geometry of eigenvectors
22.6.4
More about diagonalizable matrices
Back Matter
chevron_left
A
GNU Free Documentation License
Bibliography
Index
Colophon
Section
3.3
Terminology and notation
Vandermonde matrix
an
m
×
(
n
+
1
)
matrix where the entries in each row form a sequence
1
,
x
i
,
x
i
2
,
x
i
3
,
…
,
x
i
n
for some number
,
x
i
,
so that the full matrix has form
🔗
[
1
x
1
x
1
2
⋯
x
1
n
1
x
2
x
2
2
⋯
x
2
n
⋮
⋮
⋮
⋮
1
x
m
x
m
2
⋯
x
m
n
]
🔗
🔗
🔗
