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MTH 2011 (Linear Algebra)
MTH 2011 (Linear Algebra)
General Information:
A level 2 course in linear algebra to develop theory and methods needed in subsequent modules.
Welcome slides (Monday) and Tutorial Information slides (Thursday).
Suggested Routine:
Before Monday's lecture. Have a read over the planned lecture notes (on Canvas).
After Monday's lecture. Review* the lecture. See can you use the notes to answer questions you have.Tuesday/Wednesday. Have a go at the homework. Attend your tutorial, work with your classmates, and ask questions. Read over the planned lecture notes (on Canvas) for Thursday.
After Thursday's lecture. Review* the lecture. Submit your homework. Read over the planned lecture notes (on Canvas) for Friday.
After Friday's lecture. Review* the lecture, try to square away what you've learned this week. Have a good weekend!
*what does it mean to "Review a lecture"? Well, you probably won't have understood everything that happened in the 50 minute period. It is very normal to feel confused and a little lost when material is first presented, which is why it is so important to invest time after class to reviewing and understanding the days' material. This booklet from Loughborough University shows you how to train yourself to "self-explain", to check if you understand mathematics lectures. Usually this involves rereading notes, redoing examples without the answer before you, trying homework problems, and explaining ideas to your classmates.
Remember: learning = doing. In order to get your brain to learn something, you generally have to make it do something active. Conversely, if you do something, you are learning! Mathematics courses are designed to keep us moving through material quickly -- even if it feels like you are just treading water, that is making progress.
Our lecture notes are heavily based on the lecture notes provided by Dr David Barnes (with thanks).
Lecture 1: Linear Equations and Gaussian Elimination. Handwritten notes (Dave) and Typed notes. Updated 18/09.
Lecture 2: Backsubstitution and Matrices. Handwritten notes and Typed notes. Updated 18/09.
Lecture 3: Matrix Equations. Handwritten notes and Typed notes. Updated 19/09.
Lecture 4: Elementary Operations. Handwritten notes and Typed notes. Updated 22/09.
Lecture 5: Calculating the inverse. Handwritten notes and Typed notes. Updated 25/09.
Lecture 6: LU factorisation. Handwritten notes and Typed notes. Updated 26/09.
Lecture 7: Determinants. Handwritten notes and Typed notes. Updated 29/09.
Lecture 8: Further Properties of Determinants. Handwritten notes (correction made in purple, in proof of Proposition 8.4) and Typed notes. Updated 02/10.
Lecture 9: Fields and Vector Spaces. Handwritten notes and Typed notes. Updated 3/10.
Lecture 10: Subspaces. Handwritten notes and Typed notes. Updated 6/10.
Lecture 11: Spanning sets and linear independence. Handwritten notes and Typed notes. Updated 9/10.
Lecture 12: Bases and dimension. Handwritten notes and Typed notes. Updated 10/10. (This lecture was delivered online -- the recording can be found on Canvas, under "Announcements".)
Lecture 13: Linear complements. Handwritten notes and Typed notes. Updated 13/10.
Lecture 14: Rank of a matrix. Handwritten notes and Typed notes. Updated 16/10.
Lecture 15: The Extended Kronecker-Capelli Theorem. Handwritten notes and Typed notes. Updated 17/10.
Lecture 16: Linear maps. Handwritten notes and Typed notes. Updated 20/10.
Lecture 17: Properties of Linear maps. Handwritten notes and Typed notes. Updated 24/10.
Lecture 18: Linear isomorphisms. Handwritten notes and Typed notes. Updated 24/10.
Lecture 19: Kernel, image, and rank of a linear map. Handwritten notes and Typed notes. Updated 27/10.
Lecture 20: On the connection between matrices and linear maps. Handwritten notes and Typed notes. Updated 30/10.
Lecture 21: Matrix multiplication and composition of linear maps. Handwritten notes and Typed notes. Updated 31/10.
Lecture 22: Eigenvalues and eigenvectors. Handwritten notes and Typed notes. Updated 3/11.
Lecture 23: Examples of Eigenvalues and Eigenvectors. Handwritten notes and Typed notes. Updated 6/11.
Lecture 24: Diagonalisable linear maps. Handwritten notes and Typed notes. Updated 7/11. See information about formulas for solving polynomial equations below, as well!
Lecture 25: Inner product spaces. Handwritten notes and Typed notes. Updated 10/11.
Lecture 26: Norms, and Cauchy-Schwarz. Handwritten notes and Typed notes. Updated 13/11.
Lecture 27: Linear Isometries. Handwritten notes and Typed notes. Updated 14/11.
Lecture 28: Orthonormal sets. Handwritten notes and Typed notes. Updated 17/11.
Lecture 29: Gram-Schmidt Orthogonalization. Handwritten notes and Typed notes. Updated 20/11.
Lecture 30: Adjoints. Handwritten notes and Typed notes. Updated 21/11.
"Gaussian Elimination" is an algorithm for solving systems of linear equations, by applying "elementary row operations" to a matrix to reduce it to a simpler form (row-echelon form -- incidentally, the word echelon comes from the French word échelon, which means the "rung" or step of a ladder. Can you guess why the name applies?).
According to Wikipedia, "The algorithm that is taught in high school was named for Gauss only in the 1950s as a result of confusion over the history of the subject" -- Carl Friedrich Gauss in 1810 developed a notation for row reduction that was subsequently widely adopted, but he was not the inventor of the method! This method appears (without proof) as a calculation tool in a Chinese mathematical text approximately 2000 years ago (see the above article from the American Mathematical Society for more information). In Europe, the method was introduced by Isaac Newton in the early 1700s, as he found there was not a systematic or complete account of solving simultaneous linear equations available in the algebra textbooks of the time. According to the AMS article, the method is taught today in universities via matrices in a presentation described by the English mathematician Alan Turing, the "father" of computer science.
Eigenvalues and Eigenvectors
Eigenvalues and Eigenvectors
A linear map can be thought of as a transformation of vectors that does some combination of rotating, stretching, and shearing. To the left is a shearing transformation applied to the Mona Lisa: if every point in the image is represented by a vector pointing from the centre of the image to that point, multiplying (on the left) by a certain 2 x 2 matrix A brings us from the left to the right image.
Vectors off of the centre horizontal line such as v are sheared by A. However vectors such as v remain pointing in the same direction. v is an eigenvector of this map: it is a nonzero vector that satisfies the equation Av = λv, where λ is a scalar (in this example, it appears λ = 1).
An eigenvector v is a nonzero vector whose direction remains unchanged by the transformation v --> Av. It may be stretched/contracted, hence the equation Av = λv.
Formulas for solving polynomial equations
Formulas for solving polynomial equations
We learn the "-b" formula in school, to determine the solutions to a quadratic polynomial equation ax^2 + bx+ c = 0. There are formulas for solving a cubic (degree 3) or quartic (degree 4) equation -- which are at the end of these notes. What about a formula -- using +, x, and taking roots -- for a quintic (degree 5) polynomial equation? It has been mathematically proven that no such formula can exist! (Even stronger than this is the following specific example: the roots of the polynomial x^5 - x - 1 = 0 cannot be expressed using addition, multiplication, and taking roots.)
This discovery marked a significant moment in the history of mathematics. The original proof of this fact was due to Paolo Ruffini and Niels Abel (the history of the result is a little roundabout). A deeper and (in a sense) more general proof, that allowed one to determine which polynomials had no such formulas, was given by the French mathematician and political activist Évariste Galois (pictured right). Galois died in the midst of producing his great mathematical works -- for reasons that remain unknown, he fought in a duel and died from his wounds at the age of 20. Convinced of his impending death, Galois wrote a lengthy letter the night before the duel, outlining and collecting his mathematical ideas. This letter has been called "the most substantial piece of writing in the whole literature of mankind" (Hermann Weyl). Before his death, Galois introduced a precise connection between the roots of a polynomial and (a group of) permutations of those roots. This allows us to characterise properties of the solutions of a polynomial equation -- e.g., whether they can be expressed using +, x, and roots -- in terms of structural properties of the group of permutations. This connection is now known as Galois theory, in his honour.
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