## Linear Algebra Chapter 3 Notes

Linear Algebra Chapter 3 notes from the Strang, G.”Introduction to Linear Algebra,” 5th ed. textbook packed with important definitions, pictures, and examples. Notes are neat, compact, and color-coded for easy reference (blue for important notations, definitions (boxed in blue), and values, red for important asides and remarks)

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## Welcome To The 1806 Linear Algebra Front Page

Exams: Fridays, Feb 28, Apr 3, and May 1 all in Walker during the class hour . The final exam is 9am on Friday May 15 in JNSN-Ice Rink.

If you know you will need to miss an exam for an athletic game, please let your TA know early. Taking two courses at the same time is not an accepted excuse to miss an exam.

Grading; 15% Homeworks, 3 exams 45%, final exam 40%

Problem sets are due 11:59pm on Wednesdays through electronic submission on Gradescope. .)Collaborations allowed, but write up your own work.

The Math Learning Center is a great resource for academic help. We are hoping the level of computing won’t require much help, but we are looking into a special tutor just for this purpose.

Announcement: On Tuesday February 18, MIT will hold Monday classes. Lecture will be held, recitations will not.

You are visitor number

## Problem Set 11 Page 8

1 The combinations give a line inR

3 a plane inR

3 all ofR

2 v+w= andvw= will be the diagonals of the parallelogram with

vandwas two sides going out from.

3 This problem gives the diagonalsv+wandvwof the parallelogram and asks for

the sides: The opposite of Problem 2. In this examplev= andw= .

43 v+w= andcv+dw= .

5 u+v= andu+v+w= and 2 u+2v+w= =

. The vectorsu,v,ware in the same plane because a combination gives

. Stated another way:u=vwis in the plane ofvandw.

6 The components of everycv+dwadd to zero because the components ofvand ofw

add to zero.c= 3andd= 9give. There is no solution tocv+dw=

because3 + 3 + 6is not zero.

7 The nine combinationsc +dwithc= 0, 1 , 2 andd= will lie on a

lattice. If we took all whole numberscandd, the lattice would lie over the whole plane.

8 The other diagonal isvw. Adding diagonals gives 2 v.

9 The fourth corner can beoror. Three possible parallelograms!

10 ij= is in the base .i+j+k= is the opposite corner

from. Points in the cube have 0 x 1 , 0 y 1 , 0 z 1.

11 Four more corners,,,. The center point is(

#### 3 / 2 , 1 /2).

14 Moving the origin to 6 : 00 addsj= to every vector. So the sum of twelve vectors

changes from 0 to 12 j= .

26 Two equations come from the two components:c+ 3d= 14and 2 c+d= 8.The

solution isc= 2andd= 4. Then2 + 4 = .

27 A four-dimensional cube has 2 4 = 16corners and 2 ·4 = 8three-dimensional faces

and 24 two-dimensional faces and 32 edges in Worked Example2.4 A.

components ofv+w= andvw= . Add to find 2 v=

**Read Also: Ccl4 Molecular Geometry **

## Singular Values And Vectors : Av = U And A = U Vt

#### 7.1 Singular Vectors in *U* and *V*Singular Values in

#### 7.2 Reduced SVD / Full SVD / Construct *U V*T from *A*T*A*

#### 7.3 The Geometry of *A=U V*T : Rotate Stretch Rotate

#### 7.4 *Ak* is Closest to *A* : Principal Component Analysis PCA

#### 7.5 Computing Eigenvalues of *S* and Singular Values of *A*

#### 7.6 Computing Homework and Professor Townsend’s Advice

#### 7.7 Compressing Images by the SVD

#### 7.8 The Victory of Orthogonality : Nine Reasons

## System Of Linear Equations

Essence of Linear Algebra talks about the Matrix as a list of column vectors. In this note, we explore the Matrix as a set of linear equations . We go over Gaussian Elimination, Matrix Inversion and A = LU Factorization; the topics that were left out of Essence of Linear Algebra videos. They follow Gilbert Strangs 18.06 Course on MIT OCW. Also note, for now, we only talk about square matrices and systems where num of linear equations is equal to the number of unknowns.

**Don’t Miss: Segment Addition Postulate Practice Answer Key **

## Basic Ideas Of Linear Algebra

#### 1.1 Linear Combinations of Vectors

#### 1.2 Dot Products *v;·;w* and Lengths ||*v*|| and Angles

#### 1.3 Matrices Multiplying Vectors : *A* times *x*

#### 1.4 Column Space and Row Space of *A*

#### 1.5 Dependent and Independent Columns

#### 1.6 Matrix-Matrix Multiplication *AB*

#### 1.7 Factoring *A* into *CR* : Column rank =*r*= Row rank

#### 1.8 Rank one matrices;;;*A*= times

## Derivation Of The Decomposition

Combining the basis for the row space and the basis for the nullspace into a common matrix

to assemble a general right hand sidex=

##### [

from some set of components

c 1 c 2 c 3 c 4

##### ]T

c 1c 2c 3c 4

c 1c 2c 3c 4

Inverting the coefficient matrixAby using the teaching codeelim.mor augmentation andinversion by hand gives

##### A 1 =

So the coefficients ofc 1 ,c 2 ,c 3 , andc 4 are given by

c 1c 2c 3c 4

a+cb+dacbd

As verified by what is given in the book.

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## Unit I: Ax = B And The Four Subspaces

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 *A*x = b. We explore how the properties of *A* and b determine the solutions x and pay particular attention to the solutions to *A*x = 0. For a given matrix *A* we ask which b can be written in the form *A*x.

Looking for something specific in this course? The Resource Index compiles links to most course resources in a single page.

## Solving Linear Equations Ax = B : A Is N By N

#### 2.1 Inverse Matrices *A*-1 and Solutions *x = A*-1*b*

#### 2.2 Triangular Matrix and Back Substitution for *Ux = c*

#### 2.3 Elimination : Square *A* to Triangular *U* : *Ax = b* to *Ux = c*

#### 2.4 Row Exchanges for Nonzero Pivots : Permutation *P*

#### 2.5 Elimination with No Row Exchanges : Why is *A = LU* ?

#### 2.6 Transposes / Symmetric Matrices / Dot Products

#### 2.7 Changes in *A-1* from Changes in *A*

**Recommended Reading: Geometry Segment Addition Postulate Worksheet **

## Lectures Of Linear Algebra

These lecture notes are intended for introductory linear algebra courses, suitable for university students, programmers, data analysts, algorithmic traders and etc.

The lectures notes are loosely based on several textbooks:

**by Gilbert Strang**

*Linear Algebra and Its Applications***by David Lay**

*Linear Algebra and Its Applications***by DeFranza & Gagliardi**

*Introduction to Linear Algebra With Applications***by Gareth Williams**

*Linear Algebra With Applications*However, the crux of the course is not about proving theorems, but to demonstrate the practices and visualization of the concepts. Thus we will not engage in strictly precise deduction or notation, rather we aim to clarify the elusive concepts and thanks to Python/MATLAB, the task is much easier now.

## Eigenvalues And Eigenvectors : Ax = X And Anx = Nx

#### 6.1 Eigenvalues and Eigenvectors *x* : *Ax = x*

#### 6.2 Diagonalizing a Matrix : *X*;-1*AX* = = eigenvalues

#### 6.3 Symmetric Positive Definite Matrices : Five Tests

6.4 Solve Linear Differential Equations ; | du |

#### 6.5 Matrices in Engineering : Derivatives to Differences

#### 6.6 Rayleigh Quotients and *Sx = Mx*

#### 6.7 Derivatives of the Inverse Matrix and the Eigenvalues

#### 6.8 Interlacing Eigenvalues and Low Rank Changes in *S*

**Recommended Reading: Geometry Segment Addition Postulate Worksheet **

## Lectures And Course Notes

Lecture notes for lecture 1 – Introducing real linear systems of equations, matrices and augmented matrices, row operations, and the problems of existence and uniqueness of solutions to inear systems of equations.

Lecture notes for lecture 2 – The description of the Gauss-Jordan elimination algorithm, a complete solution to the problem of intersection of two lines in **R**2, and examples and discussion of row reduction, consistency, inconsistency, free variables, existence, and uniqueness.

Notes on vector algebra from 233 – Contains a discussion of vector arithmetic and elementary properties of vectors , the dot product, cross product, lines and planes. The notations and approach to a number of topics differ from that of this course . The problems are purely optional, and were meant to provide some challenge and practice as extra credit when I taught 233 in fall 2017. Some problems, such as in the section on dot products, will be relevant to discussions of orthogonality and inner products at the end of the course. Also, the vector spaces we refer to as **R***n* are given their own notation

Lecture notes for lecture 5 – Introducing matrix-vector products and the equation A**x**=**b**. Also contains a discussion of dot products and geometry of planes, plus two challenge problems.

Lecture notes for lectures 7 and 8 – Linear dependence and independence, with a challenge problem on the idea that linearly independent sets are minimal generating sets for spans.

## Index For Differential Equations And Linear Algebra

The index in the book Differential Equations and Linear Algebra isriddled with errors. See index errata for a fewcorrections.

I decided to re-index the whole book in parts, as part of my examreview and test prep process. Here then is a partial index

Sections indexed so far: 4.1-4.4

References

Strang, Gilbert. 18.06SC Linear Algebra, Fall 2011. ,. License: Creative Commons BY-NC-SA

**Don’t Miss: What Is The Lewis Dot Structure For Ccl4 **

## Lengths And Dot Products

The **dot product** between two vectors **v** and **w** is the number **v · w** = vw + vw ++ vw. If **v · w** = 0, **v** and **w** are perpendicular. Using Pythagoras formula, we can derive that the length of a vector v is the square root of v²+v²+v², which happens to be the square root of **v · v**. Vectors with length 1 have a special name: a **unit vector**. Some of them are i = , j = , and u = . Any non-zero vector can be turned into a unit vector in the same direction by dividing itself by its length.

Why does dot product between two perpendicular vectors turn out to be 0? We can find the proof from the *Pythagoras Law* which states that the sides of a right triangle is a² + b² = c².

The angle is less than 90° if the dot product is greater than 0, and visa versa. The dot product of two unit vectors is **cos **, revealing the exact angle between two vectors. This fact leads to the **Cosine Formula** and **Schwarz Inequality**.

## Vector Spaces And Subspaces Basis And Dimension

#### 3.1 Vector Spaces and Four Fundamental Subspaces

#### 3.2 Basis and Dimension of a Vector Space *S*

#### 3.3 Independent Columns and Rows : Bases by Elimination

#### 3.4 *Ax=*0 and *Ax=b* : ;;;*x*nullspace;;and;;*x*particular

#### 3.5 Four Fundamental Subspaces C, C, N, N

#### 3.6 Rank = Dimension of Column Space and Row Space

#### 3.7 Graphs, Incidence Matrices, and Kirchhoff’s Laws

#### 3.8 Every Matrix *A* Has a Pseudoinverse *A+*

**Also Check: Geometry Segment Addition Postulate Worksheet **

## Mit Linear Algebra Lecture : The Geometry Of Linear Equations

This is going to be my summary of Linear Algebra course from MIT. I watched the lectures of this course in the summer of last year. This was not the first time I’m learning linear algebra. I already read a couple of books and read a few tutorials a couple of years ago but it was not enough for a curious mind like mine.

The reason I am posting these mathematics lectures on my programming blog is because I believe that if you want to be a great programmer and work on the most exciting and world changing problems, then you have to know linear algebra. Larry and Sergey wouldn’t have created Google if they didn’t know linear algebra. Take a look at this publication if you don’t believe me “.” Linear algebra has also tens and hundreds of other computational applications, to name a few, data coding and compression, pattern recognition, machine learning, image processing and computer simulations.

The course contains 35 lectures. Each lecture is 40 minutes long, but you can speed them up and watch one in 20 mins. The course is taught by Gilbert Strang. He’s the world’s leading expert in linear algebra and its applications and has helped the development of Matlab mathematics software. The course is based on working out a lot of examples, there are almost no theorems or proofs.

The textbook used in this course is Introduction to Linear Algebra by Gilbert Strang.

The whole course is available at MIT’s Open Course Ware: Course 18.06, Linear Algebra.

I’ll review the first lecture today.

## What To Expect From Notes

These notes will equip you with most needed and basic knowledge for other subjects, such as Data Science, Econometrics, Mathematical Statistics, Control Theory and etc., which heavily rely on linear algebra. Please go through them patiently, you will certainly have a better grasp of the fundamental concepts of linear algebera. Then further step is to study the special matrices and their application with your domain knowledge.

**Also Check: Geometry Chapter 4 Practice Workbook Answers **

## Complex Numbers And The Fourier Matrix

#### 9.1 Complex Numbers *x+iy=rei* : Unit circle *r* = 1

#### 9.2 Complex Matrices : Hermitian *S = S*;T and Unitary *Q*-1 = *Q*;T

#### 9.3 Fourier Matrix *F* and the Discrete Fourier Transform

#### 9.4 Cyclic Convolution and the Convolution Rule

#### 9.5 FFT : The Fast Fourier Transform

#### 9.6 Cyclic Permutation *P* and Circulants *C*

#### 9.7 The Kronecker Product *A**B*