## Lecture Notes

The 14 lectures will cover the material as broken down below:

1-3: Linear Systems, Matrix Algebra

3-4: Inverses and Transposes

4-5: Vector Spaces and Subspaces

6: Bases

7: Dimension

8: Dimension and Subspaces

9-10: Linear Maps. Rank-Nullity Theorem

11-12: Matrices representing Linear Maps

13-14: Inner Product Spaces

## Lecture Notes for Linear Algebra (2021)

Publication may 2016, table of contents, preface to the notes, textbooks, websites, and video lectures, sample sections : 1.3 and 3.3 and 3.5 and 7.1.

Linear Algebra @ OCW (video lectures from MIT : Math 18.06) Linear Algebra and Learning from Data @ OCW (video lectures : Math 18.065) Gilbert Strang @ OpenCourseWare Gilbert Strang's MIT Home Page

## Other books by Gilbert Strang

Linear algebra for everyone (new textbook, september 2020), linear algebra and learning from data (2019), introduction to linear algebra, 5th edition (2016), differential equations and linear algebra, computational science and engineering, ordering gilbert strang's books, detailed table of contents, front matter (contents and preface), part 1 : basic ideas of linear algebra, 1.1 linear combinations of vectors, 1.2 dot products v · w and lengths || v || and angles θ, 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 =(1 column) times (1 row), part 2 : 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 (more advanced), part 3 : 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( a ), c( a t ), n( a ), n( a t ), 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 +, part 4 : orthogonal matrices q t = q -1 and least squares for ax = b, 4.1 orthogonality of the four subspaces (two pairs), 4.2 projections onto lines and subspaces, 4.3 least squares approximations (regression) : a t ax ̂ = a t b, 4.4 independent a 's to orthonormal q 's by gram-schmidt, 4.5 the minimum norm solution to ax = b ( n > m ) is x row space, 4.6 vector norms and matrix norms, part 5 : determinant of a square matrix, 5.1 3 by 3 and n by n determinants, 5.2 cofactors and the formula for a -1, 5.3 det ab = (det a ) (det b ) and cramer's rule, 5.4 volume of box = | determinant of edge matrix e |, part 6 : eigenvalues and eigenvectors : ax = λ x and a n x = λ n x, 6.1 eigenvalues λ and eigenvectors x : ax = λ x, 6.2 diagonalizing a matrix : x -1 ax = Λ = eigenvalues, 6.3 symmetric positive definite matrices : five tests, 6.4 solve linear differential equations du dt = au, 6.5 matrices in engineering : derivatives to differences, 6.6 rayleigh quotients and sx = λ mx (two matrices), 6.7 derivatives of the inverse matrix and the eigenvalues, 6.8 interlacing eigenvalues and low rank changes in s, part 7 : singular values and vectors : av = σ u and a = u Σ v t, 7.1 singular vectors in u and v —singular values in Σ, 7.2 reduced svd / full svd / construct u Σ v t from a t a, 7.3 the geometry of a=u Σ v t : rotate — stretch — rotate, 7.4 a k 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, part 8 : linear transformations and their matrices, 8.1 examples of linear transformations, 8.2 derivative matrix d and integral matrix d +, 8.3 basis for v and basis for y ⇒ matrix for t : v → y, part 9 : complex numbers and the fourier matrix, 9.1 complex numbers x+iy=re iθ : 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, part 10 : learning from data (deep learning with neural nets), 10.1 learning function f(x, v 0 ) : data v 0 and weights x, 10.2 playground.tensorflow.org : circle dataset, 10.3 playground.tensorflow.org : spiral dataset, 10.4 creating the architecture of deep learning, 10.5 convolutional neural nets : cnn in 1d and 2d, 10.6 counting flat pieces in the graph of f, 10.7 three-way tensors t ijk, part 11 : computing weights by gradient descent, 11.1 minimizing f(x) / solving f(x) =0, 11.2 minimizing a quadratic gives linear equations, 11.3 calculus for a function f(x, y), 11.4 minimizing the loss : stochastic gradient descent, 11.5 slow convergence with zigzag : add momentum, 11.6 direction of the step x k+1 − x k : step length c, 11.7 chain rule for ∇ f and ∇ l, part 12 : basic statistics : mean, variance, covariance, 12.1 mean and variance : actual and expected, 12.2 probability distributions : binomial, poisson, normal, 12.3 covariance matrices and joint probabilities, 12.4 three basic inequalities of statistics, 12.5 markov matrices and markov chains, 12.6 the mean and variance of z = x + y, part 13 : graphs, flows, and linear programming, 13.1 graph incidence matrix a and laplacian matrix a t a, 13.2 ohm's law combines with kirchhoff's law : a t cax = f, 13.3 max flow-min cut problem in linear programming, 13.4 linear programming and duality : max = min, 13.5 finding well-connected clusters in graphs, 13.6 completing rank one matrices.

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## Browse Course Material

Course info.

- Prof. Gilbert Strang

## Departments

- Mathematics

## As Taught In

- Linear Algebra

## Learning Resource Types

Lecture notes.

## Lecture Notes on Linear Algebra

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## Linear algebra

Unit 1: vectors and spaces, unit 2: matrix transformations, unit 3: alternate coordinate systems (bases).

## IMAGES

## VIDEO

## COMMENTS

Michaelmas M1: Linear Algebra I (2021-22) Lecture Notes Lecture Notes The 14 lectures will cover the material as broken down below: 1-3: Linear Systems, Matrix Algebra 3-4: Inverses and Transposes 4-5: Vector Spaces and Subspaces 6: Bases 7: Dimension 8: Dimension and Subspaces 9-10: Linear Maps. Rank-Nullity Theorem

Learn the fundamentals of linear algebra with these lecture notes by Cesar O. Aguilar, a professor of mathematics at Geneseo. The notes cover topics such as systems of linear equations, matrices, linear transformations, determinants, eigenvalues, and eigenvectors, with examples and exercises. The notes are based on the textbook A First Course in Linear Algebra by Lyryx Learning.

These linear algebra lecture notes are designed to be presented as twenty ve, fty minute lectures suitable for sophomores likely to use the material for applications but still requiring a solid foundation in this fundamental branch of mathematics. The main idea of the course is to emphasize the concepts

Lecture Notes for Linear Algebra (2021) Textbooks, Websites, and Video Lectures Sample Sections : 1.3 and 3.3 and 3.5 and 7.1 Linear Algebra @ OCW (video lectures from MIT : Math 18.06)Linear Algebra and Learning from Data @ OCW (video lectures : Math 18.065)Gilbert Strang @ OpenCourseWareGilbert Strang's MIT Home Page

4 CHAPTER 1. WHAT IS LINEAR ALGEBRA? 1.3 Systems of linear equations 1.3.1 Linear equations Before going on, let us reformulate the notion of a system of linear equations into the language of functions. This will also help us understand the adjective \linear" a bit better. A function fis a map f: X!Y (1.2) from a set Xto a set Y.

LINEAR ALGEBRA Lecture notes for MA 434/534 Rudi Weikard Mar 30 Apr 06 Apr 13 Apr 20 Apr 27 0 50 100 150 200 250 Version of December 1, 2020. Contents Preface iii Chapter 1. Systems of linear equations1 ... Linear Algebra may be described as the theory of nite-dimensional vector spaces. Many results, though, hold also in in nite-dimensional ...

Lecture 1: Linear Algebra Erfan Nozari October 18, 2021 Linear algebra is the most fundamental pillar of linear systems and controls. A comprehensive coverage of linear algebra can take years!, and is way beyond our scope here. In this lecture I cover only some of the basic concepts and results that we will use later in the course.

A set with a linear structure in the sense of our discussion is called a linear space or vector space. So Linear Algebra studies these linear spaces and the maps between them that are compatible with the linear structure: linear maps. This may sound somewhat abstract, and indeed, it is. However, it is exactly this level of abstraction

FUNDAMENTALS OF LINEAR ALGEBRA James B. Carrell [email protected] (July, 2005)

These are lecture notes for a ﬁrst course in linear algebra; the prerequisite is a good course in calculus. The notes are quite informal, but they have been carefully read and criticized by ... These are lecture notes for our 2-credit introductory linear algebra course. They correspond pretty closely to what I said (or should have said) in ...

Of course instructorscan produce their own lecture notes, and have done so for years, but this remains an onerous task. The publishing industry arose from the need to provide authors with copy-editing, edi- ... This text includes the basis for a two-semester course in linear algebra. • Chapters 1-4 providea standardone-semester course of 35 ...

Interactive Linear Algebra Dan Margalit and Joe Rabino have written a free online textbook called Interactive Linear Algebra, with a version speci cally created for this course. https://textbooks.math.gatech.edu/ila/1553/ The content of the course (i.e., the material that is fair game for exams) is exactly what you see in the textbook.

June 20, 2020 Basic Concepts and Notation Linear algebra provides a way of compactly representing and operating on sets of linear equations. For example, consider the following system of equations: 4x1 5x2 = 13

Math 223: Linear Algebra Lecture Notes Lior Silberman. These are rough notes for the course in winter 2022. Problem sets solutions were posted on an internal website. Contents Introduction 4 0.1. Administrivia 4 0.2. Course plan (subject to revision) 4 0.3. Change-of-language 4 Chapter 1. Vector spaces and Linear maps 5

This provides video lectures of the full linear algebra courses 18.06 and 18.06 SC and 18.065. ZoomNotesforLinearAlgebra:GilbertStrang Preface 1 ... themselves will see these notes. A happypart of linear algebra isthe wonderful variety ofmatrices—diagonal, triangular,

Lecture 1: Linear Algebra Erfan Nozari November 26, 2020 Linear algebra is the most fundamental pillar of linear systems and controls. A comprehensive coverage of linear algebra can take years!, and is way beyond our scope here. In this lecture I cover only some of the basic concepts and results that we will use later in the course.

Math121:LinearAlgebraandApplications Fall2019 13October22,2019 50 13.1UpperTriangularMatrices..... 50 14October24,2019 52

These lecture notes correspond to the course Linear Algebra II, as given at Queen Mary, University of London, in the ﬁrst sememster 2005-6. The course description reads as follows: This module is a mixture of abstract theory, with rigorous proofs, and concrete calculations with matrices. The abstract component builds

Lecture Notes | Linear Algebra | Mathematics | MIT OpenCourseWare Linear Algebra Lecture Notes pdf ZoomNotes for Linear Algebra pdf MIT18_06SCF11_Ses1.10sum.pdf pdf MIT18_06SCF11_Ses1.11sum.pdf pdf MIT18_06SCF11_Ses1.12sum.pdf pdf MIT18_06SCF11_Ses1.1sum.pdf pdf MIT18_06SCF11_Ses1.2sum.pdf pdf MIT18_06SCF11_Ses1.3sum.pdf pdf

1.3 Some More Special Matrices 1.4 Summary 2 System of Linear Equations 2.1 Introduction 2.2 Main Ideas of Linear Systems 2.3 Square Matrices and Linear Systems 2.4 Miscellaneous Exercises 2.5 Summary 3 Vector Spaces 3.1 Vector Spaces: Definition and Examples 3.2 Fundamental Subspaces Associated with a Matrix 3.3 Linear Independence

Lecture Notes for Linear Algebra (Math 203) Example Course Syllabus (from Winter 2012) Index of Terms. Matrices and Applications. Vectors. Matrices and Vectors. Matrices and Differential Equations. Practice Exams. Lecture Notes for Other Courses.

Math 223: Linear Algebra Lecture Notes Lior Silberman. These are rough notes for the course in winter 2021. Problem sets solutions were posted on an internal website. Contents Introduction 4 0.1. Administrivia 4 0.2. Course plan (subject to revision) 4 0.3. Change-of-language 4 Chapter 1. Vector spaces and Linear maps 5

Unit 1: Vectors and spaces. Vectors Linear combinations and spans Linear dependence and independence. Subspaces and the basis for a subspace Vector dot and cross products Matrices for solving systems by elimination Null space and column space.