Math 320: Linear Algebra

Fall 2020

Topics Covered in Lectures

Lecture Number	Date	Content	Corresponding Reading material
1	Oct. 05	- ℝ^2, ℝ^n, ℭ and ℭ^n 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, F^n with componentwise addition and scalar multiplication as a v.s. over F, ℭ^n 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 ℝ^2 and ℝ^3, 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: F^nand 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 F^n.	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 ℝ^nand ℭ^n as generalizations of the dot product on ℝ^2.	Pages 85-87 & 97-99 of Axler
21	Nov. 23	Basic properties of the Euclidean inner product on ℝ^nand ℭ^n, 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.