Summer 2019 MAT 224 Syllabus

--- tags: MAT224,teaching --- # Summer 2019 MAT 224 Syllabus ### Grading Scheme: 40% highest midterm+20% lowest midterm + 40% Final ### Midterms No make up exams, if not able to attend, ask doctor for doctor note and submit on quecurs before deadline. See requirements attached. **Midterm 1** on Friday 5.24. 17:00-19:00 EX200 **Midterm 2** on Friday 6.7. 17:00-19:00 EX200 **Final Exam** Between 6.19-6.26. ??? ### Class Time Code|Course Time|Location -|-|- LEC0201|Mon,Thu:**15:00-18:00**|BA1130 LEC0101|Tue,Thu:**12:00-15:00**|MP 102 ### Tutorials Code|Course Time|Location -|-|- TUT0101|Tue,Thu:**15:00-16:00**|BA2155 TUT0102|Tue,Thu:**15:00-16:00**|BA1200 TUT0201|Tue,Thu:**16:00-17:00**|BA2145 TUT0202|Tue,Thu:**16:00-17:00**|BA1230 TUT0301|Mon,Thu:**13:00-14:00**|BA2185 TUT0302|Mon,Thu:**13:00-14:00**|WB219 TUT0501|Mon,Thu:**18:00-19:00**|BA2195 TUT0502|Mon,Thu:**18:00-19:00**|WB219 ### Office Hours Name|Office Hour|Location|Email| -|-|-|- Qirui Li|Mon,Thu **11:00-12:00**;**14:00-15:00**|BA 6204|qiruili@math.utoronto.ca Ozgur Esentepe|Tue,Thu **10:30-11:45**|HU 1027|ozgur.esentepe@mail.utoronto.ca ### TA Name|Email -|- Bai, Yuguang (Roger)|rogerbai@math.utoronto.ca Gherghe, Sebastian|sebastian.gherghe@mail.utoronto.ca Girard, Vincent|vincent.girard@mail.utoronto.ca Talidou, Afroditi|atalidou@math.toronto.edu ### Piazza Link for Online Discussion https://piazza.com/utoronto.ca/summer2019/mat224h1f ### Textbook A Course in Linear Algebra David B. Damiano, John B. Little.(For reference and practicing problems) ### Prequisite Linear Algebra I and basic knowledge in Calculus. # Class Objective Our main goal for this course are following - **Part 1**: Apply linear algebra technique to **space of functions**. To do this, we introduce - Linear space of polynomials and differential operators - Some natural basis of it, Lagurange Interpolation Polynomials. - **Part 2**: A structural understanding of a **linear space with a single operator** $T$. To serve this purpose we emphasis - Invariant Subspaces and its classification - Restriction of a linear operator to invariant subspaces. - Spectral mapping theorem, eigenvalue, rank of polynomial iteration of $T$. - Young Diagram to describe dimension of $\ker T^k$ - Jordan Canonical Form - Operators commuting with $T$ and classficiation in terms of invariant subspace. - **Part 3**:Theory of Normal operators in Euclidean and Unitary spaces. Projection and its application of projection to data fitting. - Inner Product space $\mathbb{R}^n$ and unitary space $\mathbb{C}^n$. - Projection, data fitting and Grad-Smith Othogonal Process - Singular Value Decomposition - Self Adjoint Operators and Hermitian and Symmetric matrices. - Spectural Theorem. ### Extra Reading topics: The following is not on the main-logic line of class, but helpful to understand linear algebra. It serves people who have extra interest on linear algebra: - [ ]Rational canonical form - [ ] Matrix exponential $\exp(\frac{\mathrm d}{\mathrm{d}x})[f](x)=f(x+1)$ - [ ] Derivative of Matrices with one varible $\lambda$ - [ ] Derivative of determiants. - [ ] Dual space - [ ]Quotient space - [ ]Tensor Product of Vector space - [ ] Alternating product of vector space. Inner forms induced on alternating product. Relation between alternating product and subspaces and dual space and determinant. - [ ]Symmetric product of vector space. # Class Schedule ## Part 1. Linear Spaces - Class 1: Review Linear Combination（Section 1.3 Page 21） - Review of Linear Combinations - Decomposing vector as linear combinations(Linear Equation). - Matrix Multiplication(A table of linear combination). - Inverse matrix( ``` Learning Objectives: - Able to compute matrix product (Page 82:7) - Given r.r.e.f. one can find solutions directly. - Given r.r.e.f. one can tell linear combinations directly - Understand how product of matrix is a linear combination of rows or columns. - Understand how the linear combination does not change under multiplication. - Write down coordinates of the vector in vector space with given basis. - Represent a vector by basis multiply coordinates. Able to change basis by substitution of change of basis matrix. - Determine whether several vectors are linarly independent(Page 31:1 Page 46:5 Page 60:5,7,9 For problems Page 31:2,4,5,6,7,8,10 Write down strict proof and draw picture and State geometric intuition for each proof) ``` ``` Tutorial: - Obeserve r.r.e.f. by eye - Read linear combination data from r.r.e.f. - Practice using matrix product to find coefficient of linear combination - Write down solution directly by looking at r.r.e.f. - Solving unknown entry in matrix product. - Change of basis by substitution - Determine linear independece and minimal span set ``` - Class 2: Abstract Vector spaces.（See Chapter 1） - Introduction to Complex numbers $\mathbb{C}$ and its matrix realization. - Vector space $\mathbb{C}^n$. - Vector space $P^n(\mathbb{C})$: Polynomial of degree less than n. - Lagurange Polynomial: For value at certain given points - Talor Expansion: For knowing derivatives - Geometric n-dimensional space with selected origin. - Definition of linarly independence: spans the dimension as the number of vector. - Basis. - Coordinates and change of basis for Linear Transformations. - Coordinate criterion to determine independence, and span. ``` Learning Objective: - Able to prove a given set with an operation is a vector space( Page 11:6) - Able to verify wheather a given element is in the vector space( Page 19:2; Page 46,3) - [X]Able to prove some given subset is a subspace.( Page 19:2) - Able to give basis by monomials (x-a)^n or Lagrange Interpolation Polynomials (Page 58: 16. I need to make a problem. finding polynomials saitiesfies dirivative condition.) - [X]Able to describe subspace using equations or span of vectors, and translate from one to another. - Able to find a basis for the intersection of two subspaces - Able to prove linearly independence of polynomials by plug-in value method or derivative method. ``` ``` Tutorial: - Writting complex number as Ae^ix and its multiplication. - Lagurange Interpolation Polynomial - Vandemonde Matrix - A proof of Vandemonde Matrix is invertible - A suprisingly easy method of finding inverse of Vandemonde matrix - Write polynomial as linear combination of Lagurange Interpolation Polynomial or monomials ``` ``` Homework: Page 11: 6; Page 19:1,2,3,5,10; ``` - Class 3: Subspaces and direct sum of spaces - Definition of subspaces - Prove certain subset is a subspace. (Using Example of functions) - Two kinds of representation of subspaces(as Equations or span) - Direct sum of vector spaces and complements ``` Learning Objectives： - Able to find minimal span set.(Page 57:9) - Able to prove vector spans certain given subspaces: (Page 25:1,2,3) - Have geometric intuition of direct sum and able to prove simple statements.(Page 26:8,9; Page 57: 12) - Able to find a complementary subspace(Page 57: 10 b) ``` ``` Tutorial: - Show that intersection of two subspaces is also a subspace. - Determine whether a given set of vector could span the given space. - Problems of finding intersection - Problems of translating the span form into equation form. - Problem for verifying whether one subspace is included into another ``` - Class 4: Linear Transformations and Operators, - Linear Operator of $\mathbb{C}^n$ - Linear Operator of $P^n(\mathbb{C})$ - Linear Transformation: domain not equal to target. - Composition of a linear transformation - Change of basis for Linear Transformation ``` Learning Objective: - Able to find matrix of linear transformation on certain basis.(Page 73: 11 Page 81:3, 5) - Able to change basis by substitution - Able to find basis for kernel and image - Able to apply rank nullity theorem - Find a linear transformation with a given kernel. - Able to find the dimension of given subspaces (Page 56: 4; 5.) ``` ``` Turorial: - Find the matrix of the Linear Operator of taking derivative. Both on basis of Lagurange Interpolation Polynomial and basis of monomials. - Given geometric space three basis, find relation on matrix between them - Find the kernel of taking dirivative - Fine the image of taking dirivative - Find the matrix of a differential operator and its kernel and image. ``` ## Part 2: Classification of similar matrices - Class 5: Kernel and Image(See 2.1,2.2,2.3,2.4) - Injective, Surjective and invertible transfomrations - Determine basis of Kernel and Image. - Rank Nullity Theorem $\dim(\mathrm{Im} A)+\dim(\ker A)=\dim(\mathrm{Domain})$ - Class 6: Restriction of Linear Transformation. (See 2.5,2.6,2.7) - Set theoriotic example of operators - Set theoriotic nilpotent operators - Yaung Diagram - Invariant subset and restriction of operators, analogue of kernel, image and deceleration of rank. - Examples: Taking derivative or differences on functions. - Invariant subspaces - Restriction of a linear oprator on invariant subspaces - More on kernel and images: - $\ker(A|_W)=\ker A\cap W$. - $\mathrm{Im}(A|_W)=A(W)$ - Key Identity: $\mathrm{Im}(BA)=\mathrm{Im}(B|_{\mathrm{Im}(A)})$ - Key Identity: $\ker(B|_{\mathrm{Im}(A)})=\ker(A)\cap \mathrm{Im}(B)$ ``` Learning Objective: - Able to find matrix of composition linear transformation on certain basis. - Able to produce a subspace that not invariant under a linear transformation. Able to verify certain subspace is invariant subspace - Able to find the matrix of a transformation restricted on an invariant subspace. ``` ``` Tutorial: - (Proof)Inclusion with various spaces, strategy of proof and rank inequalities - Proof the intersection of invariant subspaces are invariant subspaces. - Find counter example where an invariant subpace does not have complement of invariant subspaces. - (Computation)Find the matrix of a restriction of linear transformation. - Use invariant subspace to find a basis so a lienar transformation is upper triangular. - Prove if a subspace is invariant. - If AB=BA, prove B maps invariant subspace of A to invariant subspace of A. - What is invariant subspaces for scalar matrices? - Prove the linear transformation that commute with all operators is scalling matrix. - Using the inclusion of kernel and image to prove left and right cancelation rule of injective or surjective maps. - Prove the dimension property of injective and surjective maps, using rank nullity theorem. ``` - Class 7: Nilpotency(For preparation of Jordan Canonical Form). - Definition of Nilpotency. - Set Theoriotic analogue of Nilpotency. - $Im T^n$ and $\ker T^n$ and its deceleration properties. - Decreasing and Decelerating Property of $\mathrm{rank}(A^n)$.(For preparation of proving Jordan Canonical Form) - Young Tableau for dimension losses. - Practices of finding $\mathrm{rank}(A^n)$ with given informations. - **Excercise**: If AB=BA, B invertible, nilpotency of A and of AB. - Nilpotent Young Tableau for polynomials of A - Classification of nilpotent matrices and Nilpotent Jordan Block ``` Learning Objective: - Able to Estimate the range of rank(AB) with given rank A and rank B. - Able to find exactly the rank of AB with given dimension of the intersection of ker A and Im B. - Understand rank(A^2)=rank(A) implies V=ker A+Im A - Able to use decreasing property to determine the rank of some power ``` ``` Tutorials: - A set theoriotic example of a nilpotent action and Yang Tablau of it. Example x|-> sum of its digits +1 - Given partial information of Yang Tableau, draw all possible cases of Yang Tableau. - Calculating a chain for nxn big matrix. Are also some tricky one like a tree. - Some nilpotency practice problems. Like if T^3=0, then v can not be written as a linear combination of Tv,T^2 v - Nilpotency of taking dirivatives for polynomials and its Yang Tableau. Application of polynoimals of nilpotency to determine Yang Tableau for differential operators. - Some proofs for nilpotency: If P(AB) is nilpotent, so is P(BA). - Rank inequalities ``` - Class 8: Eigenvalue and Eigenvectors(See 4.1, 4.2) - Breif Review of porperties of determinant - Characteristic Polynomial - Finding eigenvectors - Eigenvalue on invariant subspace is a subset of eigenvalues. ``` Learning Objective: - Able to justify a vector is an eigenvector(Page 173: 1) - Able to compute and understand coefficeint of characteristic polynomial for simple matrix(Page 173:2; 5) - Know how to determine eigenvalue for upper-triangular matrix(Page 173:11) - Able to find eigenvalue and eigenvectors for finite dimensional vector space(Page 173:3) - Understand Eigenspaces (Page 173:4, also show that ker(A-\lambda I) is eigenspace for \lambda. Page 175:15,17) - Able to determine algebraic multiplicity and geometric multiplicity. - Able to find eigenvalue on some given invariant subspace. ``` ``` Tutorial: - Prove the characteristic polynoimal of AB and BA are the same. - Given trace of A, A^2, A^3, determine the characteristic polynomial. - The eigenvalue and eigen vectors for P(T) where P is some polynomial. - If AB=BA the eigenvectors of A. - Calculate the char poly for 3x3 matrices ``` - Class 9: Diagonalization and Jordan Canonical Form.(See 6.3,6.4,6.5) - Eigenspaces and generalized eigenspaces. - Generalized eigenspace decomposition, Two important invariant subspaces $\ker (A-\lambda I)^\infty$ and $\mathrm{Im}(A-\lambda I)^\infty$ - Proof of Jordan Canonical Form - Young Diagram for generalized eigen spaces - Algebraic and geometric multiplicity: Definition and read from Young Diagram. - Find $P$ so $P^{-1}AP$ is a Jordan Canonical Form. - Spectral mapping theorem. - If AB=BA. then $\ker(A-\lambda I)^\infty$ is an invariant subspace of B. ``` Learning Objective: - Understand Nilpotency(Page 175:14 b: A is nilpotent if and only if all eigenvalue are 0) - Able to find JC form directly by computing the rank and using power rank.(Page 182: 1,2) - Know what the power of Jordan block looks like.(Excercise: Give a polynomial, and plug in Jordan block.) - Draw Young Diagram for genralized eigenspace for each eigenvalue. - Able to find basis to write a linear transformation into Jordan Canonical Form. - Use JCF and spectral mapping theorem to prove certain properties(Page 183: 7,8,9) - Able to draw block graph to P(A) and use it to prove statemnts(Page 182: 6) - Able to determine the JC Form for matrix p(A) where A is arbitrary polynomial. - Able to use Young Tableau to find its JCF and determine the diagonalizability and invertibility and trace and determinant. - Able to prove some cases where B could commute with A. ``` ## Part 3 Linear Operators on Unitary spaces - Class 10: Unitary Spaces(See 4.3,4.4,4.5,4.6) - Unitary spaces, subspaces and Othorgonal Complement - Projection, Data fitting. Grad-Smith orthogonalization and orthonormal basis - Class 11: Normal Operators and Spectual Theorem - Self-Adjoint operator, Symmetric and Hermitian matrices. Othogonal and Unitary operators, Normal Operators. - Spectual Theorem for Normal Operators. ``` Learning Objective: - Able to apply GS Orthogonal Process ``` ``` Tutorial: - Data fitting Problem. ``` - Class 12: Buffer ## University Schedule Day|What happen -|- May 6 |F and Y code courses begin May 9 |F and Y code course waiting lists turned off May 12 |Last day to make changes in F and Y code courses May 20 |Break June 2019|TBD June examination timetable posted on Arts & Science web site (tentative) June 4 |Last day to cancel F courses; Last day to CR/NCR in F section code courses June 14 |Last day of classes in F courses. Deadline to request Late Withdrawal (LWD) from F section code courses at College Registrar's Office June 17 |Make-up Monday June 24 |First day to request November 2019 graduation on ACORN June 19 - 26|June examination period (final exams in F courses; possible term tests for Y section code courses) # Other Information ### Class Participation Class Participation is encouraged. Another way to learn is by reading and practicing suggested problems. ### Doctor note Medical notes will be accepted ONLY from MDs with a valid CPSO number. Submit your note with a University of Toronto Verification of Student Illness or Injury form available at http://www.illnessverification.utoronto.ca/index.php Some remarks are following ``` These forms must be submitted within 3 business days of the missed test. • The form must have all required fields filled properly and legibly. • The form must be original. • The form is only considered valid if completed by a qualifed medical doctor - not an acupuncturist, chiropractor, naturopath or other health care professional. • Upon submission of the documentation review of the medical note will be done before it is accepted as valid. The review may include following up with your doctor, your college registrar, or other departmental advisors. ``` ### Athletic Absence Members of a U of T sports team which has an event on the date of tests must get a letter on University letterhead from your coach for absence. The letter must sent prior to the week of the midterm. ### Academic Integrity Cheating (including plagiarism) is very serious. Cheating can result in failure or worse. Any collusion or fabrication during or after test/quiz situations will be vigorously pursued. This includes talking (or making other extraneous noises of any kind) during a test. Every semester this situation is dealt with through administrative channels that have serious academic consequences for the student. ### Other resources **Accessibility Accommodations** http://www.studentlife.utoronto.ca **Writing and English Language Instruction** http://www.writing.utoronto.ca English Learning Language (ELL) program http://www.artsci.utoronto.ca/current/advising/ • Student Life Programs and Services: http://www.studentlife.utoronto.ca • Academic Success Centre: http://www.studentlife.utoronto.ca/asc • Health and Wellness Centre: http://www.studentlife.utoronto.ca/hwc