Linear Algebra (Math 110) - Summer 2012
University of California, Berkeley
MTuWTh 10am-11am, Room 3109 Etcheverry (Lecture)
MTuWTh 11am-12pm, Room 3109 Etcheverry (Discussion)
Contact: ![[my email]](myemail.jpg)
Office: 853 Evans Hall
Office Hours: Monday 12.10-1.40pm, Thursday 12.10-1.40pm - 853 Evans Hall
Course Outline
From the online schedule: Matrices, vector spaces, linear transformations, inner products. Determinants. Eigenvectors. QR Factorisation. Quadratic Forms and Rayleigh's Principle. Jordan Canonical Form, applications. Linear Functionals.
The aim is to cover all of these topics (except possibly Rayleigh's Principle) although not necessarily in the given order. Our main topics of study will be the Jordan Canonical Form and Euclidean and Hermitian spaces. If time permits we will consider quadratic forms on Euclidean and Hermitian spaces. For a detailed outline of the course see the syllabus (Syllabus updated 6/26).
Prerequisites: Math 54 or equivalent (Lower Division Linear Algebra). See the prerequisite knowledge sheet.
Resources
There are many textbooks on linear algebra available, each having their own merits and deficiencies. This course will not follow any particular text too closely but there will be lecture notes provided at the beginning of class and uploaded online. The 'required textbook' below has been chosen for both its merits and its (cheap) price but the content of the course will not be lifted straight from it. Rather, think of the text as an aid to the course and a supplement to the course notes.
I have also provided some recommended textbooks that I personally like and some that other maths graduate students like, as well as pointing you in the direction of the 'Linear Algebra' section of the library. I hope that these references will allow you to gain a different perspective on the material taught during class. However, the notes that I hand out and the content I write on the board during class will be what you are expected to know.
WARNING: Having a lot of different references can be both helpful and a hindrance: don't spend all your time looking through textbooks if you don't understand something, come to office hours or send me an email and I will endeavour to help you understand. Better still, speak to your classmates!
Required Textbook: Linear Algebra - Georgi E. Shilov; Dover Publications.
Recommended Textbooks:
- Linear Algebra (Second Edition) - Hoffmann, Kunze My personal favourite
- Linear Algebra (Third Edition) - Lang
- Linear Algebra (Fourth Edition) - Friedberg, Insel, Spence
- Linear Algebra Done Right - Axler Beware: this book does not introduce determinants until the end. This can make finding eigenvalues a little tricky...
- Linear Algebra and Its Applications - Strang See also MIT Open Courseware
- Finite-Dimensional Vector Spaces - Halmos
Library: Linear algebra textbooks can be found in the library with Library of Congress call numbers beginning QA184-191. Books with titles such as 'An Introduction to Linear Algebra', 'Elementary Linear Algebra' and 'Undergraduate Linear Algebra' are a pretty safe bet to include the material we will cover in this course.
The following textbooks are on reserve in the Mathematics Library in Evans Hall:
- Linear Algebra - Shilov
- Linear Algebra (Second Edition) - Hoffmann, Kunze
- Linear Algebra (Third Edition) - Lang
- Linear Algebra (Fourth Edition) - Friedberg, Insel, Spence
piazza.com: I have started a class forum at piazza.com, if you would like to added to this then please send me an email. Feel free to ask any questions you have concerning the material covered during class and on any problems you have with homework; also, please answer any questions you feel comfortable discussing with your fellow students. Try to be civil with each other!
Here is some useful information on problem solving techniques given by the Hungarian mathematician George Polya.
Here is an interesting article about the German mathematician Hermann Grassmann and his involvement in the development of linear algebra during the 19th Century.
Lecture Notes
Lecture notes for the class will be made available here intermittently throughout the course. If you find any errors (mathematical or grammatical) in the notes then please send me an email outlining the problem and I will get things sorted as soon as possible.
- The above (completed!) notes as a single file. (Updated 7/31)
Content: Here is a brief outline of what was covered during classtime. All page references refer to the lecture notes above.
- 6/18: Introduction. Basic set theory, functions. Injective/surjective/bijective functions. Fields. (p. 1-8)
- 6/19: Abstract definition of vector space. Examples: vector space of column vectors, vector spaces of functions, vector space of matrices, trivial vector space. (p. 8-14)
- 6/20: Definition of subspace, examples of subspaces, characterising subspaces in K^{n}. Sum, intersection, direct sum. Characterisation of direct sum. (Non)trivial linear relations. Linearly (in)dependent subsets. Sets that contain the zero vector are linearly dependent. (p. 14-20)
- 6/21: Characterised linearly dependent sets. Examples of linearly (in)dependent sets. General approach to determing linear (in)dependence of finite sets of column vectors. Definition of span. span as smallest subspace containing E. Elimination Lemma. Definition of linear morphism, endomorphism, isomorphism, kernel, image. Characterised injective linear morphisms. Invariance of domain. (p.20-29, Examples will be covered on Monday)
- 6/25: Examples of linear morphisms: characterised linear morphisms between K^{n} and K^{m}. Infinite dimensional example of an injective/non-surjective linear morphism. Introduced ordered sets. Definition of basis. (up to p.30)
- 6/26: Worked through worksheet problems. Defined ordered basis. A basis is a linearly independent spanning set. Any vector can be written as a unique linear combination of elements of a basis. Any basis defines an isomorphism between V and K^{n}. Defined the B-coordinate morphism. Bases exist. Basis theorem. Defined dimension, infinite dimensional spaces. Vector spaces of the same dimension are isomorphic. (p. 30-34)
- 6/27: Examples of bases, dimension. Worked through worksheet problems. Discussed how to find bases. Linearly independent sets are 'small'. Spanning sets are 'not too small'. Linearly independent sets with n vectors are a basis. Spanning sets with n vectors are a basis. Dimension of a subspace U is no greater than the dimension of V. Bases of subspaces can be extended to bases of the whole space. Dimension formula. Existence of complementary subspaces. Coordinates, change of coordinate matrix. Change of coordinate matrix is unique. (p.34-39)
- 6/28: Change of coordinates matrix. Practically determining change of coordinates matrix using standard bases. Change of coordinates matrix is invertible. Using diagrams to understand algebraic relationships. Examples of change of coordinates matrix. Matrix of a linear morphism with respect to bases. Defining property of matrix of a linear morphism. Isomorphism between sets of morphisms and matrices; properties of this isomorphism. Understanding morphisms using matrices and pivots. Injective/surjective/bijective are the same for morphisms between finite dimensional spaces of the same dimension. Composition of functions is matrix multiplication. Using morphisms to determine properties of matrices. Relating matrices with respect to different bases. Examples. (p.39-44)
- 7/2: Rooftop diagram. Using diagrams to express formulae. Similar matrices. Matrices are similar if and only if they are the matrix of the same linear morphism with respect to different bases. Definition of rank/nullity of a morphism/matrix. Rank Theorem with proof.Rank equals maximum number of linearly independent columns of a matrix. Existence of bases for which a morphism takes simple form. Two matrices are the matrix of the same linear morphism with respect to different bases if and only if they have the same rank. (p. 45-49)
- 7/3: Discussed Worksheet 6/28 problems. Introduced elementary matrices for row/colum operations. Discussed how to find P,Q used in the 'Classification of Morphisms Theorem' using elementary matrices.
- 7/5: Definition of eigenspaces, eigenvalues, eigenvectors of morphisms/matrices. Eigenvectors associated to distinct eigenvalues are linearly independent. Characteristic polynomial/equation of a morphism/matrix. Diagonalisable morphisms/matrices. How to find eigenvectors/eigenvalues. (p. 54-58)
- 7/9: Recalled definition of eigenspaces, eigenvectors, eigenvalues, diagonalisable endomorphisms/matrices. Defined algebraic/geometric multiplicity of eigenvalues. Geometric multiplicity can be no larger than algebraic multiplicity. A square matrix is diagonalisable if and only if the algebraic multiplicity of every eigenvalue equals its geometric multiplicity. An nxn matrix with n distinct eigenvalues is diagonalisable. Definition of f-invariant subspace. Every subspace is invariant with respect to scalar endomorphisms. Eigenspaces of f are f-invariant. Matrices of endomorphisms wiht respect to bases obtained by extending a basis from an invariant subspace. Block diagonal matrices. (p. 58-61)
- 7/10: Recalled criterion for diagonalisability, f-invariant subspaces, block diagonal matrices. Gave numerical criterion for diagonalisability of 2x2 matrices using trace and determinant. Block diagonal matrices. Worksheet problems. Defined nilpotent endomorphisms, matrices. Proved that an eigenvalue of a nilpotent endomorphism must be 0. Defined exponent of f, height of vectors (with respect to f), H_{k}. (p. 61-62)
- 7/11: Recalled definiions of exponent, height, H_{k}. Constructed basis for which a nilpotent endomorphism is `almost diagonal'. Defined the notion of a partition of a positive integer. Defined the partition associated to a nilpotent matrix/endomorphism. Nilpotent matrices are similar if and only if the partitions associated to them are equal. Worked through several examples to determine partitions associated to nilpotent matrices. (p. 62-66)
- 7/17: Algebraic properties of the integers. Defined the polynomial algebra in one variable over the complex numbers. Defined a representation of the polynomial algebra. Representations are completely determined by the choice of an endomorphism of V. The kernel of a representation is nonzero. A nonzero polynomial in the kernel of a representation of minimal degree divides every other polynomial in the kernel. (p. 68-70)
- 7/18: Reviewed definition of minimal polynomial and the algebraic properties of the polynomial algebra and the integers. Gave examples of Euclidean algorithm for integers and division algorithm for polynomials. Discussed how to use minimal polynomial to determine the polynomial relations satisfied by a matrix. Defined the notion of relatively prime polynomials. If f_{1},...,f_{k} are relatively prime then there exists g_{1},...,g_{k} such that f_{1}g_{1}+...+f_{k}g_{k}=1. (p. 70-73)
- 7/19: Worksheet examples. Defined annihilating polynomials. If f=gh, with g,h relatively prime then we can find a direct sum decomposition of V into L-invariant subspaces U, W such that g is an annihilating polynomial of U and h is an annihilating polynomial of W. Primary decomposition theorem. Endomorphisms are diagonalisable if and only if the minimal polynomial is a product of distinct linear factors. The roots of the minimal polynomial are the same as the roots of the characteristic polynomial. Cayley-Hamilton theorem. Discussed how to use primary decomposition to obtain Jordan canonical form. (p. 74-81)
- 7/23: Defined dual spaces, dual morphisms, dual basis. Choosing a basis defines and isomorphism between V and V*. Matrix of the dual morphism with respect to dual bases is the transpose of the matrix with respect to given bases. Relating properties of a morphism to properties of the dual morphism. Canonical isomorphism of a space with the double dual. Definition of annihilators of subspaces. Examples. (p. 49-52)
- 7/24: Definition of bilinear form, (anti-)symmetric bilinear forms. Vector space structure on set of bilinear forms. Matrix of a bilinear form. Isomorphism between Bil(V) and nxn matrices. Defining property of the matrix of a bilinear form. Change of coordinates and matrices of bilinear forms. (p. 84-88)
- 7/25: Nondegenerate bilinear forms. Matrix of nondegenerate bilinear forms are invertible. A nondegenerate bilinear form defines an isomorphism between V and V*. The matrix of this isomorphism with respect to a basis and its dual basis is just the matrix of the bilinear form. B-complements. Dimension formula for a subspace and its B-complement. (p. 88-91)
- 7/26: Definition of adjoint morphisms (with respect to B). Computing the matrix of the adjoint. Polarisation identity. If B symmetric then there exists v such that B(v,v) is nonzero. Classification of complex symmetric nondegenerate bilinear forms: for any such form there is a basis of V such that the matrix of B is the nxn identity. Classification of real symmetric nondegenerate bilinear forms: there is a basis of such that the matrix of B is diagonal, with only 1s and -1s on the diagonal. The number of 1s and the number of -1s do not depend on the basis. Computing the canonical form of a real symmetric nondegenerate bilinear form. (p. 91-99)
- 7/30: Definition of inner product, positive definite property. Definition of Euclidean space, Euclidean morphism, orthogonal transformation. Definition of the norm function, length. Examples. Triangle inequality, Pythagoras' theorem, Cauchy-Schwartz inequality. Inner products are nondegenerate. There is (essentially) one n-dimensional Euiclidean space. Any two Euclidean spaces of the same dimension are Euclidean isomorphic. (p. 99-104)
- 7/31: Worksheet problems. Orthogonal transformations, orthogonal matrices. Euclidean morphisms are injective. Euclidean morphisms between Euclidean spaces of the same dimension are isomorphisms. Orthogonal matrices have orthonormal columns/rows. Orthogonal complements. A Euclidean space is the direct sum of a subspace and its orthogonal complement. Projections onto subspaces. (p. 104-106)
Exams
Exam 1 will take place on Monday July 16th in 3109 Etcheverry Hall. The exam will start promptly at 10.15am and finish at 12pm. PLEASE ARRIVE IN CLASS AS EARLY AS POSSIBLE.
Practice Exams:
Exam 2 will take place on Thursday August 9th in 3109 Etcheverry Hall. The exam will start promptly at 10.15am and finish at 12pm. PLEASE ARRIVE IN CLASS AS EARLY AS POSSIBLE.
Here are some review notes on Jordan canonical form and bilinear forms.
Here are some review problems on Jordan canonical form and bilinear forms. Here are some solutions to the JCF problems and the bilinear forms problems.
Homework
Short Homework will be due at 10.10am in Etcheverry 3109 on Monday and Wednesday Thursday, except the first and fifth Monday (6/18, 7/16) and Wednesday July 4 Thursday August 9. All homework problem sets will be handed out in class and posted below.
Long Homework will be due at 10.10am in Etcheverry 3109 on Tuesday, except the fifth Tuesday (7/17) when Long Homework will be due Thursday (7/19).
Late homework will not be accepted.
If you are unable to submit your homework at the required time you can leave it in my mailbox (situated in the 9th floor mail room of Evans Hall, opposite the North Elevators) at any time before 10.10am on the day it is due. Please email me if you intend to leave your homework in my mailbox.
Collaboration on homework is welcome and encouraged although if you are working with another student please state that you have done so (eg. if you work with A. Nother on a particular question just write "This question was completed with A. Nother."). Failure to declare collaboration with another student will result in a grade penalty (and it is remarkably simple to tell when students have copied each other). Also, if you have used a textbook or online notes to help you understand/solve a problem please cite a reference (eg. if you used pages 52-60 of Prof. X's online lecture notes just write "This question used p.52-60 of Prof. X's online lecture notes, available at www.math.com/~profx/linalg)
Short Homework:
Long Homework:
Worksheets
Here are copies of the worksheets that are handed out during class. Please attempt these questions even if we do not get around to them during class as they will provide insight to various concepts and give you more practice at solving problems.
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