18.06SC | Fall 2011 | Undergraduate

# Linear Algebra

## Unit I: Ax = b and the Four Subspaces

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The big picture of linear algebra: Four Fundamental Subspaces.

Mathematics is a tool for describing the world around us. Linear equations give some of the simplest descriptions, and systems of linear equations are made by combining several descriptions.

In this unit we write systems of linear equations in the matrix form Ax = b. We explore how the properties of A and b determine the solutions x (if any exist) and pay particular attention to the solutions to Ax = 0. For a given matrix A we ask which b can be written in the form Ax.

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### Session Overview

 Professor Strang recommends this video from his Computational Science and Engineering I course (18.085) as an overview of the basics of linear algebra.

### Session Activities

#### Lecture Video and Summary

• Read Section 1.3 in the 4th or 5th edition.

#### Problem Solving Video

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### Session Overview

 The column space of a matrix A tells us when the equation Ax = b will have a solution x. The null space of A tells us which values of x solve the equation Ax = 0.

### Session Activities

#### Lecture Video and Summary

• Read Section 3.1 through 3.2 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 This session introduces the method of elimination, an essential tool for working with matrices. The method follows a simple algorithm. To help make sense of material presented later, we describe this algorithm in terms of matrix multiplication.

### Session Activities

#### Lecture Video and Summary

• Read Section 2.2 through 2.3 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 For this exam, you will need to know the results from elimination and the process of elimination using the simple matrices E — with one nonzero off the diagonal.

### Check Yourself

#### Exams and Solutions

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### Session Overview

 By approaching what we’ve learned from new directions, the questions in this exam review session test the depth of your understanding. Notice the short questions (with answers) at the end. This unit reached the key ideas of subspaces — a higher level of linear algebra. Please review the list of topics on the left.

### Session Activities

#### Lecture Video and Summary

• Review Chapters 1, 2, and 3, plus Section 8.2 in the 4th edition or Chapters 1, 2, and 3, plus Section 10.1 in the 5th edition.

#### Problem Solving Video

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### Session Overview

 This session explains inverses, transposes and permutation matrices. We also learn how elimination leads to a useful factorization A = LU and how hard a computer will work to invert a very large matrix.

### Session Activities

#### Lecture Video and Summary

• Read Section 2.6 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 This session explores the linear algebra of electrical networks and the Internet, and sheds light on important results in graph theory.

### Session Activities

#### Lecture Video and Summary

• Read Section 8.2 in the 4th edition or Section 10.1 in the 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 A basis is a set of vectors, as few as possible, whose combinations produce all vectors in the space. The number of basis vectors for a space equals the dimension of that space.

### Session Activities

#### Lecture Video and Summary

• Read Section 3.5 in the 4th edition or Section 3.4 in the 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 As we learned last session, vectors don’t have to be lists of numbers. In this session we explore important new vector spaces while practicing the skills we learned in the old ones. Then we begin the application of matrices to the study of networks.

### Session Activities

#### Lecture Video and Summary

• Read Section 3.3 and 8.2 in the 4th edition or Section 3.3 and 10.1 in the 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 This lecture looks at matrix multiplication from five different points of view. We then learn how to find the inverse of a matrix using elimination, and why the Gauss-Jordan method works.

### Session Activities

#### Lecture Video and Summary

• Read Section 2.4 through 2.5 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 We apply the method of elimination to all matrices, invertible or not. Counting the pivots gives us the rank of the matrix. Further simplifying the matrix puts it in reduced row echelon form R and improves our description of the null space.

### Session Activities

#### Lecture Video and Summary

• Read Section 3.2 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 We describe all solutions to Ax = b based on the free variables and special solutions encoded in the reduced form R.

### Session Activities

#### Lecture Video and Summary

• Read Section 3.3 through 3.4 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 For some vectors b the equation Ax = b has solutions and for others it does not. Some vectors x are solutions to the equation Ax = 0 and some are not. To understand these equations we study the column space, nullspace, row space and left nullspace of the matrix A.

### Session Activities

#### Lecture Video and Summary

• Read Section 3.6 in the 4th edition or Section 3.5 in the 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 A major application of linear algebra is to solving systems of linear equations. This lecture presents three ways of thinking about these systems. The “row method” focuses on the individual equations, the “column method” focuses on combining the columns, and the “matrix method” is an even more compact and powerful way of describing systems of linear equations. Portion of Fig. 2.2 from the textbook Introduction to Linear Algebra.

### Session Activities

#### Lecture Video and Summary

• Read Section 1.1, 1.2, and 2.1 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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### Session Overview

 To account for row exchanges in Gaussian elimination, we include a permutation matrix P in the factorization PA = LU. Then we learn about vector spaces and subspaces; these are central to linear algebra.

### Session Activities

#### Lecture Video and Summary

• Read Section 2.7 in the 4th or 5th edition.

### Check Yourself

#### Problems and Solutions

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## Course Info

Fall 2011
##### Learning Resource Types
Lecture Videos
Exams with Solutions
Lecture Notes
Recitation Videos
Problem Sets with Solutions
Simulations
Course Introduction
Instructor Insights