FOLLOW ALSO THE MATERIALS IN THE ONLINE EDUCATION SYSTEM OF DEU USING SAKAI, you will find there some supporting video lectures for our course:
https://online.deu.edu.tr
The documents, homeworks etc. for the course that we have given/assigned are the below files ordered by date.
Make a review the topics we have seen in the previous term; see the web page of the course MAT 2037 Linear Algebra I in the previous term.
TEXTBOOKS:
[1] Shifrin, T. and Adams, Malcolm R. Linear Algebra: A Geometric Approach. Second edition, W. H. Freeman and Company, 2011.
[2] Friedberg, S. H., Insel, A. J., and Spence, L. E. Linear Algebra. 4th edition. Pearson, 2003.
[3] Weintraub, S. H. A Guide to Advanced Linear Algebra. The Mathematical Association of America, 2011.
Lecture notes from a student's notebook from 2014 Spring (not complete) [large file, 312 MB]
Determinants - Lecture notes from a student's notebook from 2015 Autumn (not complete)
Determinants - Geometric Motivation (lecture notes)
Determinants (lecture notes)
Homework 0 - Determinants (with Worksheet 0 Problems)
Answers to Homework 0 - Determinants (in Sakai)
When studying determinants, it will be useful if you know well the sign of permutations in the symmetric group Sn, and even and odd permutations. See the notes The Sign of a Permutation by Keith Conrad (among the many nice Expository Papers).
You shall remember the geometric meaning of determinants from your multivariable calculus; in the Change of Variables Theorem for Multiple Integrals, Jacobian determinant enters the formula. For this viewpoint, see pages 88-130 for determinants from the recently published book Calculus and Analysis in Euclidean Space by Jerry Shurman whose web page also contains many nice notes which are useful for your other courses also. You shall like reading this book to learn rigorously and in a clean way main topics from advanced calculus.
The Matrix of a Linear Transformation and the Change of Basis Formula The main part of these lecture notes, the diagrams and matrices, have been typed in LaTeX by Yolay Akgün and edited by me, thanks to him.
Orthogonal Projections onto Finite-Dimensional Subspaces with a few additional exercises to Worksheet 1 Problems.
Answers to Questions 1-5 in Worksheet 1 Problems (in Sakai)
Solutions to Some Questions in Worksheet 1 Problems by Zübeyir Türkoğlu
Eigenvalues and Eigenvectors (some lecture notes)
Spectral Theorem for Symmetric Matrices with real number entries
For the classification of quadric surfaces in the 3-dimensional space R3, take a photocopy of pages 97-103 and 383-390 from the following book:
Pressley, A. Elementary Differential Geometry. 2nd Edition, Springer, 2010.
Remember graphs of quadric surfaces from your analytic geometry and calculus courses.
Answers to Questions 1, 4, 8, 9, 10 in Worksheet 2 Problems (in Sakai)
Solve all the problems in the above Worksheet 1 Problems and Worksheet 2 Problems and come to lectures prepared to discuss your answers to them.
Midterm and Final Examinations of the five years 2014-2018 and 2022
You can find answers to some of these examinations from the faculty photocopy room but I strongly advice you to work hardly to solve them before seeing the answers. You cannot be a problem solver by reading answers to problems. As a preparation for your midterm, solve the five midterms in the above set of examinations, you must be able to solve each of them in at most three hours, make yourself these examinations. It will be a good preparation if you can solve all of these five midterms and understand all the reasoning needed to answer them. See also the above Subjects of the Midterm.
Your textbooks has nice problems to develop your understanding and solving them will make you prepared fully for the midterm, you must learn well studying your textbooks, they are nice textbooks to learn basics of linear algebra.
Midterm Examination
Answers to the Midterm Examination
Study vector spaces over the field C of complex numbers and inner product spaces over the field C of complex numbers, see the lecture notes.
Complex Vector Spaces and Complex Inner Product Spaces
Complex Eigenvalues and the Jordan Canonical Form
Worksheet 3 Problems. Study Section 7.1 of your textbook; this is the final section that we shall see from your textbook.
Answers to Worksheet 3 Problems (in Sakai)
For a better understanding of the Jordan canonical form and more examples, see either Chapter 7 Canonical Forms from the second textbook by Friedberg, Insel and Spence, or see the following short supplementary textbook:
Weintraub, S. H. Jordan Canonical Form, Theory and Practice. Morgan & Claypool, 2009.
In the remaining weeks of the term, for the topics we shall study, study mainly Chapters 6 and 7 from the following second textbook, take a photocopy of the sections that we shall study:
Friedberg, S. H., Insel, A. J., and Spence, L. E. Linear Algebra. 4th edition. Pearson, 2003.
Some of the topics that we shall study till the end of the term are the following topics; see the Syllabus:
- Inner Product Spaces over the field C of complex numbers.
- Orthogonal projection onto a finite-dimensional subspace of an inner product space over R or C.
- The adjoint T*: V → V of a linear operator T : V → V on an inner product space V over R or C.
- T-cyclic subspaces, Cayley-Hamilton Theorem, minimal polynomials of square matrices and characterizing diagonalizability using the minimal polynomial.
- Dual Space V* that consists of all linear functionals on the vector space V over the field F, that is, linear maps from V to the field F.
- Self-adjoint (=Hermitian), Unitary and Normal Matrices and Operators.
- Schur's Theorem.
- Spectral Theorem for Normal Matrices and Operators, The Spectral Theorem in terms of projection maps.
- Symmetric bilinear forms and quadratic forms; Nonorthogonal diagonalization of quadratic forms and Slyvester’s Law of Inertia; Positive definite and positive semidefinite operators and matrices, Slyvester’s Criteria for Positivity.
Worksheet 4 Problems about T-cyclic subspaces, Cayley-Hamilton Theorem, minimal polynomials of square matrices and characterizing diagonalizability using the minimal polynomial.
To solve the Worksheet 4 Problems, use the following summary of lecture notes:
The T-cyclic subspaces, the Cayley-Hamilton Theorem and the minimal polynomial (see lecture notes for the proofs)
Answers to Worksheet 4 Problems (in Sakai)
The topics for our last weeks are: Dual Spaces, Self-adjoint(=Hermitian), Unitary and Normal Matrices and Operators, Positive definite matrices, Nonorthogonal diagonalization of quadratic forms and Slyvester’s Law of Inertia, ...
To solve Questions 1-5 for dual spaces in the below Worksheet 5 Problems, use the following summary of lecture notes:
Dual Spaces (see lecture notes for the proofs)
To solve Questions 6-15 in the below Worksheet 5 Problems, use the following notes:
With some proofs added, the proof of the Spectral Theorem for Normal Matrices - Self-adjoint(=Hermitian), Unitary and Normal Matrices and Operators, and The Spectral Theorem in terms of projection maps
To solve Question 16 for nonorthogonal diagonalization in the below Worksheet 5 Problems, use the following summary of lecture notes:
Positive Definite Matrices and Operators, Nonorthogonal Diagonalization and Slyvester's Law of Inertia, Quadratic Forms and Bilinear Forms (see lecture notes for the proofs)
Answers to Worksheet 5 Problems (in Sakai)