Explore Linear Algebra Course Details and Benefits
Delve into the intriguing world of Linear Algebra with this comprehensive course covering key topics such as matrices, vectors, systems of equations, determinants, eigenvalues, eigenvectors, and more. Understand the advantages and disadvantages of linear concepts, and discover how algebra plays a crucial role in various fields including machine learning, cryptography, and circuit theory. Uncover the significance of linear algebra in mathematics, science, and engineering applications. Dive deeper into the course material and enhance your understanding of this fundamental branch of mathematics.
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Course Information Instructor: ; E-mail: hcyen@ntu.edu.tw Time: 9:10-12:00, Friday Place: BL 103 Personal Website: https://homepage.ntu.edu.tw/~hcyen/index.html Course Materials: Textbook + Slides ( ) Class web page: the following and NTUCool https://homepage.ntu.edu.tw/~hcyen/courses/LA-2024.html : https://sites.google.com/view/linearalgebra2024fall TA: 606 D07921012@ntu.edu.tw
Required Textbook Elementary Linear Algebra - A Matrix Approach 2nd Ed., by L. E. Spence, A. J. Insel and S. H. Friedberg ( )
Course Outline Chapter 0. Introduction Chapter 1. Matrices, Vectors, and Systems of Linear Equations Chapter 2. Matrices and Linear Transformations Chapter 3. Determinants Chapter 4. Subspaces and Their Properties Chapter 5. Eigenvalues, Eigenvectors, and Diagonalization Chapter 6. Orthogonality Chapter 7. Vector Spaces
Grading (35%) (35%) / (30%) ( )
Linear Algebra Linear: What is Linear ? Have to do with line/plane/etc Advantage: Simple Efficient algorithms Disadvantage: Too simple, (sometimes) not realistic. Near linear if we zoom in Transistor: a non-linear device Apply bias to operate in the linear region :
Algebra (Wikipedia) Algebra is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Boolean Algebra Abstract Algebra Linear Algebra Algebraic Boolean Number Theory Boolean Process Algebra Boolean Tensor Algebra
Why Study Algebra? Algebra provides a unified way of studying many problems in math., science, and engineering. E.g. Linear Algebra ( ): Machine learning, Quantum computing, Coding theory, Circuit theory, Optimization, Control theory, Boolean algebra ( ): Computer circuits, Computer programming, Formal verification, Abstract algebra ( ): Cryptography, Coding theory, Communication, Symmetry,
In this Course, Linear Algebra is about Solve the matrix equation Many practical problems can be captured by the above matrix equation. Solve system of linear equations using matrices, row reduction, inverses, etc. Solve the matrix equation ?? = ?? Solve eigenvalue problem using characteristic polynomial. Understand dynamic of linear transformations via eigenvalues, eigenvectors, diagonalization. In fact, large classes of engineering problems, no matter how huge, can be reduced to linear algebra.
What to learn What to learn in Linear Algebra? in Linear Algebra? Linear System
A system has input and output (function, transformation, operator) System Speech Recognition System How are you Dialogue System (e.g. Siri, Alexa) How are you I am fine Communication System Hello Hello
Linear System 1. Persevering Multiplication Linear System Linear System ? ? 2. Persevering Addition Linear System + + Linear System Linear System
Linear System When the input and output are vectors 1. Persevering Multiplication 3 4 5 3 4 5 Linear System 1 2 ?1 Linear System ? 2 ? = 2 6 8 10 2 4 2. Persevering Addition 3 4 5 7 8 9 7 8 9 3 4 5 1 2 Linear System + 1 2+5 6 Linear System 5 6 Linear System 10 12 14 6 8
Are they Are they Linear Linear? ?
Linear? NO x2 x System 1. Persevering Multiplication k2x2 kx2 x2 x kx System System 2. Persevering Addition ?1 2 ?1 System 2 ?1+ ?2 ?1+ ?2 System 2 ?2 ?2 System 2+ ?2 2 ?1
6 8 9 9 0 2 Linear? 6 9 8 0 9 2 System Transpose ? ? YES 6? 8? 9? 9? 0 2? 6? 9? 8? 0 9? 2? System 6 8 9 9 0 2 System 6 9 8 0 9 2 7 13 5 8 10 12 System 1 2 3 4 5 6 System 7 13 10 5 12 8 1 4 2 5 3 6
Linear? YES function f function f Derivative ? ? ? ? x2 2x ?? ?? ? + ? ? + ? 3 3x
Linear? 1 3?3 ?3 ? ? ? = ?2 Integral (from a to b) ? ? ?? Function ? ? ? scalar Area a b
? ? ? ?? ? ? Linear? YES ? ? Persevering Multiplication ?? ? ?? ? ?? ? ? = ? ? ? ?? ? ? ? Persevering Addition ? ? ? ? ?? ? ? ?? ? ? ? ? ? ? ? + ? ? ?? ? ? + ? ? ? ? ? = ? ? ?? + ? ? ?? ? ?
Input: voltage source, current source Linear output: voltage and current on the load ( ) Circuit + v i - ( ) input output Linear System
An Application Kirchhoff s Laws 10 - 8i1 -2(i1 i2 ) = 0, 12 -2(i2 i1) 7i2 3i2 = 0 - + 10i1 2i2 =10 2 i1 12i2 = -12 i1 i2 ?1 ?2 10 12 ? =10 2 12 ? = ? = 2
( ) ?(3)+ ?2?(2)+?1?(1)+ ?0? ? ??? ?(1)= ? = 0, ?(3)= ? ?(2)= ? ?1 ?2 ?3 ?1 ?2 ?3 0 0 1 0 0 1 Let = ?1= ? ?2= ?(1) ?3= ?(2) d/dt ?0 ?1 ?2 Linear Algebra meets Differential Equation x = Ax Then ? 1= ?2 ? 2= ?3 ? 3= ?(3)= ?2?(2) ?1?1 ?0?
( ) Complex but linear Basically just change of basis frequency time Fourier Transform Linear System
Why Taking One Semester? Most of you know how to solve systems of linear equations in high school ( ) Why bother spending a whole semester learning Linear Algebra? Reason: Often engineers need to solve lots of equations with lots of variables Can we gain useful information about the set of solutions without having to solve the equations? It is also challenging to recognize problems in real-world that can be formulated using linear algebra.
How to do well in this course? Come to the class and pay attention to lectures Read the textbook and class notes Do homework and exercises carefully Ask questions Discuss with your classmates (when allowed) Do not be afraid of mathematical proofs Practice, practice, practice!
