Linear Algebra

Introduction

  • Everybody knows that the world of mathematics is governed by equations. The easiest of these equations are the linear equations – ones in which the power of each variable is not greater than one, for example \mathbf{x + y = 1} , etc.
  • Many a times, mathematics is all about converting difficult (non-linear) equations into simple (linear) equations.
  • Lastly, in real world problems, you are rarely asked to solve single equations. You are required to solve system of (more than one) equations. Linear algebra is the study of how to solve such systems of linear equations.

The textbook for this course is Linear Algebra and its applications by David C. Lay (4th edition).

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Lecture 1

Overview

In this lecture, you will be introduced to the concept of system of linear equations. You will learn how to solve them. Take home lesson is that a system of linear equation has either no solution, exactly one solution or infinitely many solutions (think of linear equations as straight lines).

Activities

  • Lecture Video: watch the video Lecture 1.
  • Suggested Reading: Read section 1.1 in the textbook.

Check Yourself

Work the following problems on your own: 12, 14, 17, 21, 22, 25. Answers to odd-numbered exercises can be found at the back of the book.

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 Lecture 2

Overview

In this lecture, you will learn algorithms which will make the task of solving linear systems easy. Take home lesson is that a free variable is one which is not expressible as a relation containing no other variable. That is, it does not dependent on any other variable. Hence it is ‘free’ to take any value. Point to ponder: Is free variable unique?

Activities

  • Lecture Video: watch the video Lecture 2.
  • Suggested Reading: Read section 1.2 in the textbook.

Check Yourself

Work the following problems on your own: 13, 14, 20, 33. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 3

Overview

In this lecture, you will learn about one of the most important objects in Mathematics and other sciences — vector. You will learn how to add and subtract vectors. You will also learn how to multiply and divide a vector by a scalar (any real number). Take home lesson is that the purpose of algebraic properties of \mathbf{\mathbb{R}^n} is that since we are working with two completely different objects, namely, vectors and scalars, we need to make sure that their interaction together is mathematically consistent.

Activities

  • Lecture Video: watch the video Lecture 3.
  • Suggested Reading: Read section 1.3 in the textbook.

Check Yourself

Work the following problems on your own: 21, 22, 26. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 4

Overview

In this lecture, you will learn how to use matrices to solve system of linear equations. Once again the purpose is to make the calculations easier. Take home lesson 1 is that there are equivalent ways of solving problems in linear algebra. Either use a matrix equation, a vector equation or solve the system of linear equations algebraically. Choose the method which is easiest in the given situation. Take home lesson 2 is that what do we mean by span? Well, two vectors always define a plane. The ‘area’ (or points, regions) covered by that plane (formed out of those two vectors) is given by the notion of span.

Activities

  • Lecture Video: watch the video Lecture 4.
  • Suggested Reading: Read section 1.4 in the textbook.

Check Yourself

Work the following problems on your own: 21, 27, 36, 37. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 5

Overview

In this lesson, we will study the solution sets of linear systems. Linear systems can be categorised into two main branches: homogeneous and Non-homogeneous. Take home lesson is that by homogeneous we mean that all the variables in a linear equation have same degree. For example either all are of degree one, as in x_1 + x_2 + x_3 = 0 or {x_1}^{2} + {x_2}^{2} + {x_3}^{2} = 0 . But x_1 + x_2 + x_3 = 5  is not homogeneous equation because 5= 5 x^0 , that means in a single linear equation, we have two different types of degrees, namely 1 and 0.

Activities

  • Lecture Video: watch the video Lecture 5.
  • Suggested Reading: Read section 1.5 in the textbook.

Check Yourself

Work the following problems on your own: 5, 12, 17, 28. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 6

Overview

In this lecture, you will be introduced to the significant notion of linear independence of a set of vectors. Take home lesson is that while solving linear systems, the simplest of the solutions is the trivial solution (i.e., in case of consistent systems, set all variables equal to zero and that is a solution). But are there some non-trivial solutions as well or the trivial solution is the only solution, is an important question to answer.

Activities

  • Lecture Video: watch the video Lecture 6.
  • Suggested Reading: Read section 1.7 in the textbook.

Check Yourself

Work the following problem on your own: 31. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 7

Overview

In this lecture, you will be introduced to linear transformations. As the name suggests, these transformations modify the vectors while preserving linearity. Take home lesson 1 is that while this concept is very simple, it has wide applications. It is a great geometric tool and you are strongly suggested to play with it by applying various transformations on vectors and planes and see what they change into. For example, there are transformations via which you can kill a line and convert it into a point (try to find such transformation). Take home lesson 2 is the idea of projection. Notice that how simply you can project (see reflection) of a 3D object into 2D plane, etc.

Activities

  • Lecture Video: watch the video Lecture 7.
  • Suggested Reading: Read section 1.8 in the textbook.

Check Yourself

Work the following problems on your own: 4, 13, 15, 23. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 8

Overview

In this lecture, we will learn theorems and techniques that will help us carry out linear transformations. Take home lesson 1 is that in this lecture we will see the linearity condition in action while finding standard matrix of a transformation. Take home lesson 2 is that you will be introduced to the notions of one-one and onto transformations. It is very important that you grasp these concepts now. You should know what they mean and what they do not mean. They will be with for the rest of your life. As a motivation, try to apply the ideas of function, one-one and onto relations to the relationship of marriage.

Activities

  • Lecture Video: watch the video Lecture 8.
  • Suggested Reading: Read section 1.9 in the textbook.

Check Yourself

Work the following problems on your own: 4, 10, 18, 20. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 9

Overview

Applications.

Activities

  • Lecture Video: watch the video Lecture 9.
  • Suggested Reading: Read section 1.6 in the textbook.

Check Yourself

see lecture slides.

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Lecture 10

Overview

Revision.

Activities

  • Lecture Video: watch the video Lecture 10.
  • Suggested Reading: none.

Check Yourself

see lecture slides.

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Lecture 11

Overview

Take home lesson 1 is that each column of AB is a linear combination of the columns of A using the weights from the corresponding column of B.
Take home lesson 2 is the an invertible matrix is row equivalent to I_n.

Activities

  • Lecture Video: watch the video Lecture 11.
  • Suggested Reading: Read sections 2.1 and 2.2 in the textbook.

Check Yourself

Work the following problems on your own: 7, 31, 23. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 12

Overview

In this lesson, I would like you to pay special attention to what is meant by the terms ‘disjoint classes’ and ‘equivalent statements’. You are also encouraged to see example 2 (p. 117) for a profound application of partitioned matrices, and example 4 (p. 119) for insight into matrix multiplication.
Those who are keen to see applications of mathematics to real world will find an accessible yet remarkable example in the last sub-heading of this section.

Activities

  • Lecture Video: watch the video Lecture 12.
  • Suggested Reading: Read sections 2.3, 2.4 and 2.5 in the textbook.

Check Yourself

Work the following problems on your own: 1, 4. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 13

Overview

In this lesson, you would be introduced to the notion of a subspace. This simple notion is a foundation stone in the edifice of linear algebra. When a vector space can be constructed out of a finite set of vectors, then set of such vectors is called Spanning set. You will learn that a spanning set is a subspace of the corresponding vector space. After completing the lecture you should be able to answer that is the set containing only zero vector in \mathbb{R}^n a subspace?
Notice the significance of definitions, you will not be able to grasp this section if you do not remember when is a linear system consistent (consistent linear system: A linear system with at least one solution).
You will be introduced to two other extremely important subspaces called column and null spaces. Example 6 and 7 are especially insightful for understanding of Null and Column spaces respectively.
Take home lesson is why do we require a basis? Because (sub)spaces usually contain an infinite number of vectors. Thus it is desirable to find a rather smaller set and work with that.
Take home lesson 2 is that in general row operations drastically alter the original matrix. But you have to learn the things that remain unaltered during the process. For example, the linear dependence relations among the columns are preserved during row operations.

Activities

  • Lecture Video: watch the video Lecture 13.
  • Suggested Reading: Read sections 2.8 in the textbook.

Check Yourself

Work the following problems on your own: 8, 14. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 14

Overview

In this lecture, you will be introduced to the notions of basis, isomorphism, dimension and rank. They are building blocks of edifice of Mathematics. They are one of those glimpses of higher mathematics that you can see while still being an undergraduate. One of the take home lesson is that the spanning set is not unique, while a basis is.
Pay special attention to the notion of coordinate vector. You should think over it, to see how we are not restricted to use only one coordinate system. But can create new coordinates out of old ones. The purpose behind such creation is, as you will study in the examples in the text, is to work in coordinates that make the particular problem easier to deal with.

Activities

  • Lecture Video: watch the video Lecture 14.
  • Suggested Reading: Read section 2.9 in the textbook.

Check Yourself

Work the following problems on your own: 2, 10, 13. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 15

Overview

In this lecture, you will be introduced to the notion of determinants. Though the idea is apparently simple, it requires a lot of practice to feel easy with computation part of it. You will also study properties of determinants, and relations between value of determinant and invertibility of a matrix. This lecture effectively proves the point that there are methods that are computationally hard and thus not used in computer algorithms. Pay special attention to dynamics of a determinant of a matrix as various row operations are performed on the matrix.
This lecture coincides with the beginning of the new chapter (Chapter 3) of the book. The introduction of this chapter is a must-read!

Activities

  • Lecture Video: watch the video Lecture 15.
  • Suggested Reading: Read section 3.1 and 3.2 in the textbook.

Check Yourself

Work the following problems on your own: (section 3.2) 15, 19, 21, 23, 24. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 16

Overview

In this lecture, you will learn Cramer’s Rule for solving a system of equations. You will learn to compute the inverse of a matrix using this rule. There are alternatives to this method, but the significance of this method is timeless. You will also be introduced to very important Laplace transform. The geometric subtleties associated with determinants are also taught in detail.

 

Activities

  • Lecture Video: watch the video Lecture 16.
  • Suggested Reading: Read section 3.3 in the textbook.

Check Yourself

Work the following problems on your own: 21, 27, 31. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 17

Overview

Activities

  • Lecture Video: watch the video Lecture 17.
  • Suggested Reading: Read section X.X in the textbook.

Check Yourself

Work the following problems on your own: X. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 18

Overview

In this lecture, you will be introduced to the the ideas of a Vector Space and Subspace. You may think as the entry point into advanced Linear Algebra. The topic of subspace is further explored. Subspace enjoys several important properties and they are discussed with good examples. A take home message is that Span is a subspace. Example 11 is illustrative. Also Example 12 should ring some bells 🙂

Activities

  • Lecture Video: watch the video Lecture 18.
  • Suggested Reading: Read section 4.1 in the textbook.

Check Yourself

Work the following problems on your own: 1 (Practice Problem), 3, 9. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 19

Overview

This lecture will begin with very significant ideas of null space and column space. A beautiful contrast between null space and column space properties is presented. Linear transformations are discussed with emphasis on their range and kernel. The notion of kernel is a must-know. A take home lesson is that the solutions of a differential equation form a vector space. Try to feel the unifying nature of mathematics!

Activities

  • Lecture Video: watch the video Lecture 19.
  • Suggested Reading: Read section 4.2 in the textbook.

Check Yourself

Work the following problems on your own: 2 (practice problem), 11, 12. Answers to odd-numbered exercises can be found at the back of the book.

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Lecture 20

Overview

This lecture will discuss the notions of linear independence and dependence. They are of central importance in Linear Algebra. The very important concept of basis will be introduced. They can provide coordinates with the blessing of being unique. You will also learn about spanning set and basis of null and column spaces. Realise that this lecture is a continuation of Lecture 19. A take home lesson is that row operations can change the column space data of a matrix!
Section titled Two views of basis is a nice read.

Activities

  • Lecture Video: watch the video Lecture 20.
  • Suggested Reading: Read section 4.3 in the textbook.

Check Yourself

Work the following problems on your own: 1, 2, 3, 6, 11, 12, 26. Answers to odd-numbered exercises can be found at the back of the book.

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Undraping the Universe | Science and Mathematics of my times.

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