| Margaret E. M. Thomas | Department of Mathematics and Statistics |
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. (This course is NOT on Avenue to Learn. Please do not go looking for it there or ask me why you cannot find it there!)....News pertaining to this section of MATH 1ZC3 will appear here; for news about the course in general, it is your responsibility to check the Course Website regularly.
9 April 2019: The notes from Lecture 36 are now available online. Don't forget to fill out a Course Evaluation or to check the Course Website for review sessions and extra office hours. Good luck everyone!
4 April 2019: The notes from Lecture 35 are now available online. Also: please do send in suggestions for the review time that we have next week (e.g. topics to go over, examples of questions). For the TV reference that I made at the end of class (concerning the 75 times table), see Countdown Numbers Game . This does not usually happen. Play online yourself here . This has nothing whatsoever to do with anything (except for saying that knowledge of certain times tables can be useful in certain unlikely circumstances and therefore, very tenuously, it connects to Hill ciphers).
3 April 2019: The notes from Lecture 34 are now available online, as well as the solution to the Exercise from the end of the Lecture 33 notes.
1 April 2019: The notes from Lecture 33 are now available online, as well as the solution to the Exercise from the end of the Lecture 32 notes.
29 March 2019: The notes from Lecture 32 are now available online. Also: Course Evaluations are now open! Please take a few minutes to give your feedback on this course -- if you want to give feedback from the post-it note exercise (either reiterating those comments, or following up on your earlier comments), this would be a really helpful place to do that.
27 March 2019: The notes from Lecture 31 are now available online.
25 March 2019: The notes from Lecture 30 are now available online. Good luck tonight!
21 March 2019: The notes from Lecture 29 are now available online.
20 March 2019: The notes from Lecture 28 are now available online. Please try the Exercise from the very end of class before next time!
Also, don't forget that there are Test 2 review sessions this week:
- Thursday March 21st, 4:30pm-6:20pm in TSH/120
- Friday March 22nd, 4:30pm-6:20pm in BSB/147
Just go along to one of them at a time that suits you.
18 March 2019: The notes from Lecture 27 are now available online (including some extra workings from the class, at the end).
14 March 2019: The notes from Lecture 26 are now available online. Since we had a big of a technological malfunction during the middle of class, the punchline, the "Test for Subspaces", is written out again more explicitly at the end of the document, but we'll go over this again in class on Monday. There are also a couple of proofs of Facts about vector spaces, working with the Axioms. If you want to see any more details of the examples that we went through in class, either take a good look at the textbook, or write me (MATH 1ZC3 in the subject line).
Also, information about Test 2 is now available on the Course Website .
13 March 2019: The notes from Lecture 25 are now available online, together with a worked example (at the end of the file) showing why $M_{m,n}(\mathbb{R})$, the collection of all $m \times n$ matrices, is a vector space (working through all ten axioms).
11 March 2019: The notes from Lecture 24 are now available online.
8 March 2019: The notes from Lectures 21, 22 and 23 are now available online. If there are any topics where you want to see more worked examples, please shoot me an email (just remember to put MATH 1ZC3 in the subject line).
28 February 2019: The notes from Lectures 18, 19 and 20 are now available online. Also: Test 1 results are now out; please read the MSAF FAQ and/or the Post Test FAQ on the Course Website before contacting me about any issues discussed therein (and then please remember to include the course code MATH 1B03 in the subject line of the email if it turns out that you do still need to contact me). If you want to go over any of the mathematics, then by all means come talk to me!
19 February 2019: A typo has been corrected in the example on "powers of diagonalizable matrices" at the start of the notes in Lecture 17 (so a new version of these notes has been posted here). The typo did not appear when that example was originally given at the end of Lecture 16, and I'm really sorry if this caused any confusion. Please be careful with these formulas! It's unfortunately possible to get mixed up even after a long time, so please be aware of the issue -- you'll be OK if you pay attention and take as much care as you possibly can. I was in such a hurry to get the notes posted for you that I didn't notice this. Please let me know if you're still confused and have questions!
15 February 2019: The notes from Lectures 16 and 17 are now available online. You're welcome to email me with questions over the break, though it may take a day or two to reply. Don't forget that I'll be holding office hours on Monday 25 February, 11:30-12:30, the day of Test 1.
7 February 2019: The notes from Lectures 13, 14 and 15 are now available online.
31 January 2019: The notes from Lecture 12 are now available online.
30 January 2019: The notes from Lecture 10 provided by Dr Rusworth as well as the notes from Lecture 11 are now available online.
26 January 2019: During the coming week there will be a few changes to the scheduled programming. The class on Monday (28 January) will go ahead as scheduled but will be taken by Dr Rushworth. Office hours on Monday 28 January have been moved to Wednesday 30 January at the usual time (11:30-12:30).
24 January 2019: Lecture notes for Lectures 6-9 are now posted online.
16 January 2019:
11 January 2019: Lecture notes for Week 1 (Lectures 1-3) now posted online.
10 January 2019: Office hour for Thursday 17 January (deadline day for Assignment 1) is moved to Wednesday 16 January at the same time of 10:30-11:30. I will also not necessarily be available by email on Thursday 17 January. Make sure you plan ahead.
1 January 2019: This page up and running.
Summaries of lectures and notes from class will be posted here as the semester progresses. Lecture notes will typically be posted all together once a week, but may be posted earlier if certain topics relate to an assignment with an earlier deadline.
The lecture notes below are typically augmented versions of what was presented on screen during class; the augmentations are principally to be found in the form of extra comments (and corrections) written in light blue. Please send me an email if you ever find any mistakes or there are glitches in the documents.
You are strongly advised to go through these notes alongside your own notes carefully to make sure that you understand everything that was discussed!
(Mon 7 January 2019) -- Introduction to course, including introduction to the Course Website . 1.1 Systems of Linear Equations. (Tues 8 January 2019) -- More on 1.1 Systems of Linear Equations, 1.2 (preliminaries towards) Gaussian Elimination. (Thurs 10 January 2019) -- More on 1.2 Gaussian Elimination (and Gauss--Jordan Elimination). (Mon 14 January 2019) -- End of 1.2 Gaussian Elimination (and Gauss--Jordan Elimination); 1.3 Matrices (definition of matrix, dimensions, transpose, trace, matrix addition, scalar multiplication of matrices, matrix multiplication). (Tues 15 January 2019) -- More on 1.3 Matrices (more on matrix multiplication); 1.4 Properties of Matrices (further properties of matrix operations, identity matrices). (Thurs 17 January 2019) -- More on 1.4 Properties of Matrices (more on identity matrices, inverse matrices); 1.5: Elementary matrices and a method for finding the inverse (if it exists). (Thanks to Dr Valeriote for these notes.) (Mon 21 January 2019) -- More on 1.5: Elementary matrices and a method for finding the inverse (if it exists). (Tues 22 January 2019) -- 1.6: More on Linear Systems and Invertible Matrices. (Thurs 24 January 2019) -- More on 1.6: More on Linear Systems and Invertible Matrices; Matrix Polynomials; 1.7: Special kinds of square matrices: Diagonal (definition, sums, scalar multiples, products, inverses, powers), Triangular (definition of upper triangular and lower triangular). (start at "Triangular matrices" on the first page) and here for 2.1 (up until the first example on the last page). (Tues 29 January 2019) -- More on 2.1: Determinants by Cofactor Expansion (various tricks); 2.2: Evaluating determinants by Row Reduction (effect of row operations on determinants, determinants of elementary matrices). (Thurs 31 January 2019) -- More on 2.2: Evaluating determinants by Row Reduction (examples, Elementary Column Operations); 2.3 Properties of determinants (determinants of matrix products, inverses, scalar multiples; problems with addition and determinants). (Mon 4 February 2019) -- More on 2.3 Properties of determinants (inverses using cofactors, augmented matrices). 5.1: Eigenvalues and eigenvectors (definitions, characteristic equation). (Tues 5 February 2019) -- More on 5.1: Eigenvalues and eigenvectors (characteristic polynomial, method to find eigenvectors, eigenspaces, bases, eigenvalues for triangular and diagonal matrices). (Thurs 7 February 2019) -- More on 5.1: Eigenvalues and eigenvectors (geometric multiplicity and algebraic multiplicity); 5.2: Diagonalization (using eigenvalues and eigenvectors, if multiplicities match). (Mon 11 February 2019) -- More on 5.2: Diagonalization (similar matrices, how to diagonalize a matrix) *** UP TO THIS POINT IS THE SYLLABUS FOR MIDTERM TEST 1. *** More on 5.2: Diagonalization (finding powers of diagonalizable matrices). (Thurs 14 February 2019) -- 5.4: Differential Equations (how to solve an exponential growth/decay equation, how to solve a system of such equations when the matrix of coefficients is diagonalizable). For this topic please also see the videos "Systems of DE's Modules" the link to which is on the Course Website . (Mon 25 February 2019) -- 10.1 (from 9th Edition): Complex Numbers (complex plane, addition and multiplication). 10.2 (from 9th Edition): Division of Complex Numbers (complex conjugate, modulus). (Chapter 10 from the 9th Edition of the textbook can be found on the Course Website .) (Tues 26 February 2019) -- More on 10.2 (from 9th Edition): Division of Complex Numbers (division, inverses). 10.3 (from 9th Edition): Polar Form of a Complex Number (radius, argument, representation with trigonometric functions and the exponential function, multiplication and division). (Thurs 28 February 2019) -- More on 10.3 (from 9th Edition): Polar Form of a Complex Number (finding powers (De Moivre's Rule) and roots). 3.1: Vectors in 2-space, 3-space and n-space (reconciliation of line segments in 2-space and 3-space with row vectors/column vectors/tuples/points). (Mon 4 March 2019) -- More on 3.1: Vectors in 2-space, 3-space and n-space (basic operations in n-space, linear combinations). 3.2: Norm, Dot Product and Distance in $\mathbb{R}^n$ (norm). (Tues 5 March 2019) -- More on 3.2: Norm, Dot Product and Distance in $\mathbb{R}^n$ (unit vectors, normalization, standard unit vectors, distance, dot product, matrix notation, relationship between norm and dot product, angle, Cauchy-Schwarz Inequality, Triangle Inequality, Parallelogram Equation). (Thurs 7 March 2019) -- 3.3: Orthogonality (orthogonal vectors, normals to lines in $\mathbb{R}^2$ and planes in $\mathbb{R}^3$, Projection Theorem, orthogonal projection of a vector onto another vector, shortest distance to a line or a plane). 3.4: The Geometry of Linear Systems (parametric equations for lines in $\mathbb{R}^2$ and $\mathbb{R}^3$). (Mon 11 March 2019) -- More on 3.4: The Geometry of Linear Systems (parametric equations for lines and planes in $\mathbb{R}^n$, geometrical interpretation of solutions to $A\bar{x} = \bar{0}$ and $A\bar{x} = \bar{b}$). 3.5: Cross products (definitions, equations of planes, some properties). (Tues 12 March 2019) -- More on 3.5: Cross products (areas of parallelograms and triangles, and volumes of parallelipipeds). 4.1: Real Vector Spaces (axioms, first examples). (Thurs 14 March 2019) -- 4.1: Real Vector Spaces (review of axioms, lots of examples and non-examples, key facts). 4.2 Subspaces (definition, test for subspaces). (Mon 18 March 2019) -- More on 4.2: Subspaces (lots of examples, spans -- see extra worked examples at the end). (Tues 19 March 2019) -- More on 4.2: Subspaces (null spaces, eigenspaces). 4.3: Linear Independence (definition, some examples). (Thurs 21 March 2019) -- More on 4.3: Linear Independence (more examples, Wronskians, facts about linear independence). 4.4: Coordinates and bases (introductory remarks). (Mon 25 March 2019) -- More on 4.4: Coordinates and bases (definitions, examples). (Tues 26 March 2019) -- 6.3: Gram--Schmidt Process (orthogonal/orthonormal sets and bases, linear independence of orthogonal sets, Gram--Schmidt Process, example). (Thurs 28 March 2019) -- More on 6.3: Gram--Schmidt Process (more examples, orthogonal projection onto subspaces). 4.5: Dimension (definitions, first theorem about linear independence and spanning). Here is the solution to the Exercise set in the posted notes after the lecture. (Mon 1 April 2019) -- More on 4.5: Dimension (more theorems about linear independence and spanning, and a test for bases; examples of basis checking, finding bases). Here is the solution to the Exercise set in the posted notes after the lecture. (Tues 2 April 2019) -- 4.7 Row Space, Column Space and Null Space. (Thurs 4 April 2019) --10.14 Cryptography (encoding with Hill 2-ciphers, modular arithmetic). (Mon 8 April 2019) -- More on 10.14: Cryptography (reciprocals, inverses modulo 26, decoding with Hill 2-ciphers).