Linear Algebra 5th 3.2 Solutions

Table of Contents for Introduction to Linear Algebra (5th edition 2016)

• 1 Introduction to Vectors
  • 1.1 Vectors and Linear Combinations
  • 1.2 Lengths and Dot Products
  • 1.3 Matrices
• 2 Solving Linear Equations
  • 2.1 Vectors and Linear Equations
  • 2.2 The Idea of Elimination
  • 2.3 Elimination Using Matrices
  • 2.4 Rules for Matrix Operations
  • 2.5 Inverse Matrices
  • 2.6 Elimination = Factorization: A = LU
  • 2.7 Transposes and Permutations
• 3 Vector Spaces and Subspaces
  • 3.1 Spaces of Vectors
  • 3.2 The Nullspace of A: Solving Ax = 0 and Rx = 0
  • 3.3 The Complete Solution to Ax = b
  • 3.4 Independence, Basis and Dimension
  • 3.5 Dimensions of the Four Subspaces
• 4 Orthogonality
  • 4.1 Orthogonality of the Four Subspaces
  • 4.2 Projections
  • 4.3 Least Squares Approximations
  • 4.4 Orthonormal Bases and Gram-Schmidt
• 5 Determinants
  • 5.1 The Properties of Determinants
  • 5.2 Permutations and Cofactors
  • 5.3 Cramer's Rule, Inverses, and Volumes
• 6 Eigenvalues and Eigenvectors
  • 6.1 Introduction to Eigenvalues
  • 6.2 Diagonalizing a Matrix
  • 6.3 Systems of Differential Equations
  • 6.4 Symmetric Matrices
  • 6.5 Positive Definite Matrices
• 7 The Singular Value Decomposition (SVD)
  • 7.1 Image Processing by Linear Algebra
  • 7.2 Bases and Matrices in the SVD
  • 7.3 Principal Component Analysis (PCA by the SVD)
  • 7.4 The Geometry of the SVD
• 8 Linear Transformations
  • 8.1 The Idea of a Linear Transformation
  • 8.2 The Matrix of a Linear Transformation
  • 8.3 The Search for a Good Basis
• 9 Complex Vectors and Matrices
  • 9.1 Complex Numbers
  • 9.2 Hermitian and Unitary Matrices
  • 9.3 The Fast Fourier Transform
• 10 Applications
  • 10.1 Graphs and Networks
  • 10.2 Matrices in Engineering
  • 10.3 Markov Matrices, Population, and Economics
  • 10.4 Linear Programming
  • 10.5 Fourier Series: Linear Algebra for Functions
  • 10.6 Computer Graphics
  • 10.7 Linear Algebra for Cryptography
• 11 Numerical Linear Algebra
  • 11.1 Gaussian Elimination in Practice
  • 11.2 Norms and Condition Numbers
  • 11.3 Iterative Methods and Preconditioners
• 12 Linear Algebra in Probability & Statistics
  • 12.1 Mean, Variance, and Probability
  • 12.2 Covariance Matrices and Joint Probabilities
  • 12.3 Multivariate Gaussian andWeighted Least Squares
• Matrix Factorizations
• Index
• Six Great Theorems / Linear Algebra in a Nutshell

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Each section of the book has a Problem Set.

Linear Algebra Animation Videos

In the following videos, click the 'Play' ► icon
While playing, click the word 'YouTube'
to watch a larger video in a separate tab

Linear transformations of a house

Eigenvalues don't quite meet

Linear Algebra Problems in Lemma

My friend Pavel Grinfeld at Drexel has sent me a collection of interesting problems -- mostly elementary but each one with a small twist. These are part of his larger teaching site called LEM.MA and he built the page http://lem.ma/LAProb/especially for this website linked to the 5th edition.

The H.264 Video Standard (promised in Section 7.1 of the book)

This video standard describes a system for encoding and decoding (a "Codec") that engineers have defined for applications like High Definition TV. It is not expected that you will know the meaning of every word -- your book author does not know either. The point is to see an important example of a "standard" that is created by an industry after years of development--- so all companies will know what coding system their products must be consistent with.

The words "motion compensation" refer to a way to estimate each video image from the previous one. The simplest would be to guess that successive video images are the same. Then we would only need the changes between frames -- hopefully small. But if the camera is following the action, the whole scene will shift slightly and need correction. A better idea is to see which way the scene is moving and build that change into the next scene. This is MOTION COMPENSATION. In fact the motion is allowed to be different on different parts of the screen.

It is ideas like this -- easy to talk about but taking years of effort to perfect -- that make video technology and other technologies possible and successful. Engineers do their job. I hope these links give an idea of the detail needed.

• http://www.h264info.com/h264.html
• http://en.wikipedia.org/wiki/H.264/MPEG-4_AVC
• http://www.axis.com/files/whitepaper/wp_h264_31669_en_0803_lo.pdf

Our recent textbook Linear Algebra for Everyone starts with the idea of independent columns

    This leads to a factorization A = CR where C contains those independent columns from A

    The matrix R tells how to combine those columns of C to produce all columns of A

    Then Section 3.2 explains how to solve Rx = 0. This gives the nullspace of A  !!

Here is that new section : A = CR and Computing the Nullspace by Elimination

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Linear Algebra 5th 3.2 Solutions

Source: https://math.mit.edu/~gs/linearalgebra/

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