Math 220 - Linear Algebra

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For the time being, disregard the exam contents found on this page. It will be updated in due time.
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  • Tentative course schedule * (Fall 2013)
  • Midterm I **  
  • [1999] [2002] [2004] [2006]
  • Midterm II **  
  • [1999] [2002] [2004] [2006]
  • Final
  • [1997] [1997] [1999] [2002]
  • Exam rules and terms
  • Suggested problems   [NEW]

  • * This schedule is tentative. Depending on the class and/or requests of other related departments, I reserve the right to skip/fast forward certain topics in order to pay more attention to certain others
    ** Exam dates can be changed provided the request is filled in advance and majority of the class supports it
    Exam contents listed below are tentative and are subject to change (depending on the actual pace of the class)

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    Informal course contents

  • Linear systems and matrices. We start with a well known subject: linear system of equations. In order to formalize/simplify the process of solving such a system, we introduce the notion of matrix (which at this stage is merely dropping the unknowns and writing down only the coefficients). We study the properties/operations/forms of matrices relevant to linear systems.
  • Vector spaces. Next, we observe that dealing with matrices involves linear operations only, i.e., addition and multiplication by numbers. In order to formalize this concept, we introduce the notion of linear (vector) space. This notion is crutial for modern mathematics and its application. A great number of objects studied in math (vectors, matrices, polynomials, functions) form, in fact, vector spaces, and the solution of virtually any problem starts with an attempt to linearize it, i.e., reduce it to a question on a vector space. Alas, most vector spaces involved are infinite dimensional; we confine ourselves to finite dimensional spaces and conclude that their structure is fairly simple. The important notions introduced here are: vector space, basis, dimension, coordinates.
  • Inner product spaces. Now, we would like to study various metric properties (length, angle, volume) in vector spaces. To this end, we introduce the notion of inner product, which generalizes the well known scalar product of vectors. It is important to keep in mind that, in general, inner product is not something given by God; it is an additional structure on a vector space that should be picked and fixed in advance. Again, we conclude that the structure of a finite dimensional vector space supplied with an inner product is very simple.
  • Linear transformations. Linear transformations are maps between vector spaces preserving the two linear operations. We discover that linear transformations can be described by matrices. (In fact, it is this concept that is the true idea behind the notion of matrix.) The matrix describing a linear transformation depends on the choice of a basis. Thus, we make an attempt to discover those true intrinsic properties of a matrix (= transformation) that do not change with a basis. The notion of linear transformation is difficult to overestimate: it covers such diverse objects as linear systems, (systems of) linear differential/integral equations, derivative, integral, Fourier transform, Laplace transform, etc. *
  • Determinats. Determinant of a square matrix is nothing but the (signed, higher dimensional) volume of the (higher dimensional) parallelepiped spanned by the vectors represented by the columns (or rows) of the matrix. ** We will see that it has lots of nice properties, in particular, some sort of linearity, and learn how to calculate it.
  • Eigenvalues, eigenvectors. This is probably the most advance topic in linear algebra (and the most nonlinear one). The eigenvalues of a matrix (or linear transformation) determine its "shape". They are basis independent, and for a good matrix they encode all basis independent information about the transformation. Correct me if I'm wrong, but, to my understanding, modern quantum physics reduces to the study of eigenvalues of certain (unfortunately, discontinuous infinite dimensional...) linear transformations.
    * By the way, the success of calculus of derivatives is explained by the fact that it replaces an arbitrary function  f  by its linearization  df  and studies the resulting linear transformation!
    ** This explains the sophisticated formula of change of variables in multiple integrals. The Jacobi matrix of a transformation is its linearization; hence, its determinant (Jacobian) is the coefficient that shows how the transformation changes volumes (locally). Since integral is basically a weighed sum of volumes, we should multiply by this coefficient: dx dy dz being the old volume, the new volume is |Jdu dv dw !

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    Midterm I    (30%)   October 26, 2013, @ ?:?? am    
    See:   [
    important remarks]   [samples]   [room assignment]   (updated 10/07/2010)

    Topics covered (tentative): (updated 10/07/2010)
  • Systems of linear equations (definitions, related matrices, solving *, relation between homogeneous and non-homogeneous systems)
  • Matrices (basic concepts, matrix operations and their properties, elementary transformations and elementary matrices, equivalence of matrices, echelon forms, Gaussian elimination, nonsingular matrices and the inverse matrix *, criteria of invertibility)
  • Matrix transformations (basic concepts, examples)
  • Determinants: simple properties; poly-linearity; behavior under elementary transformations; row and column expansion; evaluation of determinants**; relation to singularity of a matrix; inverse of a matrix; application to solving linear systems; geometric applications

    Chapters (9th edition):   1 (1-6), 2 (1-3), 3 (1-5)
    * I do not request a particular way of solving a problem. However, keep in mind that certain ways may be much easier than certain others :)
    ** Please, do not use any self-invented "rules" for determinants of order greater than 3! Use elementary transformation (do not forget the fact that an extra coefficient may pop up) and row/column expansion.

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    Midterm II    (30%)   December 14, 2013, ?:?? am     See:   [important remarks]   [samples] (check also MT I)   [room assignment]   (updated 23/11/2010)

    Topics covered (tentative)*:
  • Vector spaces (basic definitions and properties; examples **)
  • Subspaces (definitions, properties, examples, detecting a subspace, defining a subspace via homogeneous linear conditions vs. as a span, the solution space of a homogeneous system, finding its basis = solving the system ***)
  • Linear combinations, linear independence, bases and isomorphisms (concepts, detecting if a vector belongs to a given span and if a given set of vectors is linearly independent***)
  • Bases and isomorphisms (concepts, basis, coordinates, isomorphism to Rn , dimension, extending a given set of vectors to a basis ***, finding a basis in a given span ***)
  • Vector (sub-)spaces related to a matrix (row space, column space, null space), their properties and relations, rank and nullity, existence/uniqueness criteria for linear systems in terms of ranks
  • Inner products and inner product spaces (concept, properties, examples ††)
  • The matrix of an inner product; positive definite matrices (if covered in class) †††
  • The Cauchy-Swarz inequality and its applications. Length (norm), distance, angle. Triangle inequality

    Chapters (9th edition):   4 (1--9), 5 (1--3)
    * The material covered in Midterm I is fully included
    ** Primary examples are Rn = Rn , Pn , and P. All other notation (including various function and matrix spaces) will be explained
    *** Note that, so far, (almost) any problem that one may encounter, no matter how it is stated, reduces eventually to one of these three (basis in the solution space, basis in a span, extending a given set to a basis). These, in turn, reduce to converting an appropriate matrix to an echelon form. So, linear algebra is made easy! One of the problems (per test) may require extra thinking
    † This is your primary tool: any problem can be reduced to a problem in Rn and, hence, to a problem about matrices. Please, before solving a problem, think about an appropriate basis. A wrong choice may transform a few lines long solution to a two pages long one (see remarks)!
    †† Primary examples are the standard inner product in Rn  and the "integral" inner product on the space of continuous functions and its subspaces
    ††† We did not treat this notion in details. However, you are supposed to be familiar with the concept and be able to determine whether a given matrix is positive definite (e.g., by completing squares)

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    Final    (30%)   TBA      [important remarks]   [samples]   [room assignment (SAZ18, SAZ20)]   (updated 1/3/2011)

    Topics covered (tentative)*:
  • Orthogonality. Properties of orthogonal (orthonormal) sets. Gram-Schmitd process
  • Orthogonal complement (notion, properties, finding). Orthogonal projection. Distance between a vector and a subspace. Orthogonality of the subspaces related to a matrix. "Quadratic"** minimization problems
  • Linear transformations: examples; kernel and range; matrix of a linear transformation
  • Linear transformations (continued): change of bases; similarity
  • Diagonalization****; eigenvalues and eigenvectors; characteristic polynomial and its invariance under similarity; diagonalizability of symmetric matrices (self-adjoint operators)

    Chapters (9th edition):   5 (1-6), 6 (1-3, 5), 7 (1-3)
    * The material covered in Midterm I and Midterm II is fully included and footnote*** applies :)
    ** I.e., those reduced to finding the orthogonal projection
    **** From last quiz I conclude that this topic might cause certain problems. So, I tried to compile a brief overview.

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    Homeworks     (10%)   There will be 5 homeworks, 2% each, assigned electronically.

    All questions regarding the homeworks are to be directed to the assistant. The assistant is instructed to  give no credit  to identical papers. The same applies to the exams.
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    Most exam problems will be taken from the textbook. Solve them in advance, and you will do well! Same concerns quizzes.

    During the exams please keep in mind the following:
  • Calculators are  not  allowed 
  • Identical solutions (especially identically wrong ones) will  not  get credit. I reserve the right to decide what "identical" means. You still have the right to complain 
  • Do not argue about the distribution of the credits among different parts of a problem. I only accept complaints concerning my misunderstanding/misreadung your solution 
  • Show all your work. Correct answers without sufficient explanation might  not  get full credit 
  • Indicate clearly and unambiguously your final result. In proofs, state explicitly each claim 
  • Do not misread the questions or skip parts thereof. If you did, do not complain 
  • If you believe that a problem is misstated, do not try to solve it; explain your point of view instead. However, do not take advantage of this option: usually problems are stated correctly! 
  • Each problem has a reasonably short solution. If your calculation gets completely out of hand, something must be wrong (e.g., you might have chosen a wrong basis) 

    Grading policy
    I will take off a few (2-3) points for arithmetical mistakes. However,  a lot  of points will be taken off for `obvious' mistakes, i.e., either those that you can easily avoid or those showing a deep misunderstanding of the subject. This includes, but is not limited to, the following:
  • Wrong dimension in a physical problem
  • Things that don't make sense
  • Inverse matrix with a row/column full of "0"s
  • Basis with repeated vectors or zero vectors
    (As it is impossible to predict every single mistake that the students can make, I reserve the right to decide what is `obvious' on the fly.)
    Furthermore, solving a different problem (other than stated), even if perfect, will give you  no credit 
    * Of course, this only applies to problems that I am grading personally
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    Last update: November 18, 2009