Math 320: Linear Algebra
Fall 2020
Topics Covered in Lectures
Lecture Number |
Date |
Content |
Corresponding Reading material |
1 |
Oct. 05 |
- ℝ2, ℝn, ℭ and ℭn properties of componentwise
addition and scalar multiplication, the use of F to denote the set of real or
complex numbers, the definition of a vector space (v.s.) over F, the vector
space of sequences F∞. |
Pages 1-10 of the textbook
(Axler) |
2 |
Oct. 07 |
Examples
of vector spaces: Trivial vector space, Fn with componentwise
addition and scalar multiplication as a v.s. over F, ℭn with componentwise addition and
scalar multiplication as a v.s. over ℝ, the vector space C of functions f: ℝ→ℭ with domain ℝ. |
Pages 1-10 of the textbook |
3 |
Oct. 09 |
Properties
of vector spaces: Uniqueness of the zero vector and additive inverse of a
vector, 0.v=0, a.0=0, (-1).v=-v; subspaces of a v.s., trivial subspace,
examples of subspaces of ℝ2 and ℝ3, set of polynomials p: ℝ→ℭ as a subspace of the function
space C. |
Pages 11-14 of the textbook |
4 |
Oct. 12 |
Sum and
direct sum of subspaces and their basic properties |
Pages 14-17 of Axler |
5 |
Oct. 14 |
A
characterization of a vector space that is the direct sum of two subspaces,
linear combinations, span of a subset, spanning subsets, finite- and
infinite-dimensional vector spaces |
Pages 17-23 of Axler |
6 |
Oct. 16 |
List of
vectors, linear dependence, a property of linear dependent lists of vectors
(Linear dependence lemma) |
Pages 23-25 of Axler |
7 |
Oct. 19 |
Theorem:
Spanning list cannot be shorter than linearly-independent lists, Proposition:
Subspaces of finite-dimensional vector spaces are finite-dimensional, basis
of a vector space, examples: Fn
and F[x], Proposition: Bases
of finite-dimensional vector spaces are finite sets. |
Pages 25-27 of Axler |
Quiz 1 |
Oct. 20 |
|
|
8 |
Oct. 21 |
Reduction of
spanning lists to linearly-independent lists, existence of bases for
finite-dimensional vector spaces, the extension of linearly independent lists
to bases, direct-sum decomposition of finite-dimensional vector spaces |
Pages 27-31 of Axler |
9 |
Oct. 23 |
The
dimension of a finite-dimensional vector space, dimension as the minimum of
the length of spanning lists and as the maximum of a linearly-independent
list, the dimension of the sum of two subspaces of a finite-dimensional
vector space. |
Pages 31-34 of Axler |
10 |
Oct. 26 |
Dimension
criteria for direct sum decomposition of a finite-dimensional vector space;
Linear operators, zero and identity operators, other examples of linear
operators, the set of linear operators T:V->W with domain V, dual space of
a vector space, characterization of the elements of the dual space of Fn. |
Pages 34-39 of Axler |
11 |
Oct. 30 |
The vector space of everywhere-defined
linear operators L (V,W), composition of linear operators, the image and inverse image
of subspaces under a linear operator, the range and null space of a linear
operator, the characterization of one-to-one linear operators in terms of
their null space. |
Pages 40-44 of Axler |
12 |
Nov. 02 |
Dimension theorem, linear equations and the
existence and uniqueness of their solution. |
Pages 45-47 of Axler |
13 |
Nov. 04 |
Invertible linear operators (vector-space
isomorphisms), the uniqueness and linearity of the inverse of an invertible
operator, isomorphic vector spaces, the classification problem for vector
spaces, vector-space structures |
Pages 53-55 of Axler |
14 |
Nov. 06 |
Classification of the finite-dimensional vector
spaces by their dimension, invertible linear operators mapping a
finite-dimensional vector space onto itself, vector space of m x n matrices |
Pages 55-58 of Axler |
15 |
Nov. 09 |
Matrix representation of elements of a
finite-dimensional vector space in a given basis, matrix representation of linear
operators mapping a finite-dimensional vector space to another
finite-dimensional vector space, multiplication of an m x n matrix by an n x
1 matrix |
Pages 47-53 of Axler |
Quiz 2 |
Nov. 10 |
|
|
16 |
Nov. 11 |
Composition of linear operators and multiplication
of matrices, block-diagonal matrix representations of a linear operator
mapping a finite-dimensional vector space to the same vector space |
Pages 47-53 & 75-76 of
Axler |
17 |
Nov. 13 |
Invariant subspace of a linear operator T:V→V,
One-dimensional invariant subspaces of T, eigenvectors, eigenvalues, and the
point spectrum of T, invariant subspaces associated with the eigenvalues of T |
Pages 76-78 of Axler |
18 |
Nov. 16 |
Diagonal matrix representations of linear operators
acting in a finite-dimensional vector space; results leading to the proof of
the existence theorem for eigenvalues of a linear operator acting in a
finite-dimensional complex vector space. |
Pages 87-90 & 79-81 of
Axler |
Quiz 3 |
Nov. 17 |
|
|
19 |
Nov. 18 |
Upper-triangular matrix representations of linear
operators acting in a finite-dimensional vector space |
Pages 81-85 of Axler |
20 |
Nov. 20 |
Invertibility for linear operators acting in a
finite-dimensional vector space and having an upper-triangular matrix representation
in some basis, eigenvalues as diagonal entries of upper-triangular matrix
representations of an operator acting in a finite-dimensional vector space
(if such a rep. exists); Euclidean inner product on ℝn and ℭn as
generalizations of the dot product on ℝ2. |
Pages 85-87 & 97-99 of
Axler |
21 |
Nov. 23 |
Basic properties of the Euclidean inner product on ℝn and ℭn, inner
products on a vector space and inner-product spaces, some examples of inner
product spaces, norm of vectors, unit and orthogonal vectors, orthonormal
subsets of an inner-product space |
Pages 100-102 of Axler |
22 |
Nov. 25 |
Pythagorean theorem, Cauchy-Schwarz and triangular inequalities,
properties of the norm defined by the inner product, the notion of a norm on
a vector space and normed spaces, the metric defined by the norm, the notion
of a metric on a nonempty set and metric spaces, norm of linear combination
of elements of an orthonormal subset of an inner-product space, the linear
independence of orthonormal subsets of an inner-product space |
Pages 102-107 of Axler |
23 |
Nov. 27 |
Orthonormal bases of an inner-product space, matrix
representations of elements of a finite-dimensional inner-product space in an
orthonormal basis, Gram-Schmidt orthogonalization, existence of orthonormal
basis for a finite-dimensional inner-product space, extending orthonormal
lists to orthonormal bases of a finite-dimensional inner-product space |
Pages 107-110 of Axler |
24 |
Nov. 30 |
Upper-triangular representations of a linear
operator in an orthonormal basis of a finite-dimensional inner-product space;
Projection operators, direct-sum decomposition of a vector space in term of
the null space and range of a projection operator. |
- |
Quiz 4 |
Dec. 01 |
|
|
25 |
Dec. 02 |
Orthogonal complement of a subspace of an
inner-product space, orthogonal direct sum decompositions, and orthogonal
projections |
Pages 111-112 of Axler |
26 |
Dec. 04 |
Some basic properties of orthogonal projections (in
finite- and infinite-dimensional inner-product spaces), matrix representation
of linear operators in orthonormal bases |
Pages 113-116 of Axler |
27 |
Dec. 07 |
Dual space of a vector space, Riesz
Lemma for finite-dimensional inner-product spaces, adjoint of a linear operator
mapping a finite-dimensional inner-product spaces into another, matrix
representation of the adjoint of a linear operator in orthonormal bases |
Pages 117-118 & 121 of
Axler |
28 |
Dec. 09 |
Properties of the adjoint of linear operators,
self-adjoint operators and their matrix representations in orthonormal bases,
lemmas leading to the polarization formula for complex inner-product spaces
and characterization of self-adjoint operators T in terms of the reality of
<Tv,v> for all elements v of the inner-product
space |
Pages 119-120 & 128-129 of Axler |
29 |
Dec. 11 |
Polarization formula and realness of <Tv,v> for a self-adjoint operator acting in a real
inner-product space (proofs to be done by the students), Realness of the
eigenvalues of self-adjoint operators, commutator of two linear operators,
normal operators and their characterization in terms of their representation
in orthonormal bases, characterization of normal operators in terms of the
equality of the norms of Tv and T*v, eigenvectors and eigenvalues of normal
operators. |
Pages 130-132 of Axler |
Winter
Break |
|
|
|
30 |
Dec. 21 |
Spectral theorem for normal operators T acting in a finite-dimensional
complex inner-product space, orthogonal direct-sum decomposition of such an
inner-product space into subspaces of the form null(T-lj I) where lj are distinct eigenvalues of T |
Pages 132-134 & 137 of
Axler |
Quiz 5 |
Dec. 22 |
|
|
31 |
Dec. 23 |
Orthogonal projection operators Pj
onto null(T-lj I) for
a normal operator T acting in a finite-dimensional complex inner-product
space, the resolution of identity, orthogonality property of Pj, and the spectral resolution (expansion) of
T; Positive operators and their basic properties |
Pages 144-147 of Axler |
32 |
Dec. 25 |
Isometries and unitary operators, sets of isometries
of one dimensional real and complex Euclidean space and three-dimensional
real Euclidean space, characterization of isometries (unitary operators)
acting in a finite-dimensional inner-product space and equality of their
inverse and adjoint, eigenvalues of isometries, Spectral theorem for
isometries of in a finite-dimensional complex inner-product space, functions
of a normal operator acting in a finite-dimensional complex inner-product
space, characterization of isometries of a finite-dimensional complex
inner-product space in terms of the exponential of i
times self-adjoint operators |
Pages 147-150 of Axler |
33 |
Dec. 28 |
Polar and singular-value decompositions of a linear operator
acting in a finite-dimensional inner-product space, changes of basis and
similarity transformations, generalized eigenvectors |
Pages 152-164 of Axler |
34 |
Jan. 04 |
|
|
Quiz 6 |
Jan. 05 |
|
|
35 |
Jan. 06 |
|
|
36 |
Jan. 08 |
|
|
Note: The pages from the
textbook listed above may not include some of the material covered in the
lectures.