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Math 211 Activities Workbook:
Adapted from: Understanding Linear Algebra by David Austin
David Austin, Erica Bernstein
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\(\newcommand{\avec}{{\mathbf a}} \newcommand{\bvec}{{\mathbf b}} \newcommand{\cvec}{{\mathbf c}} \newcommand{\dvec}{{\mathbf d}} \newcommand{\dtil}{\widetilde{\mathbf d}} \newcommand{\evec}{{\mathbf e}} \newcommand{\fvec}{{\mathbf f}} \newcommand{\nvec}{{\mathbf n}} \newcommand{\pvec}{{\mathbf p}} \newcommand{\qvec}{{\mathbf q}} \newcommand{\svec}{{\mathbf s}} \newcommand{\tvec}{{\mathbf t}} \newcommand{\uvec}{{\mathbf u}} \newcommand{\vvec}{{\mathbf v}} \newcommand{\wvec}{{\mathbf w}} \newcommand{\xvec}{{\mathbf x}} \newcommand{\yvec}{{\mathbf y}} \newcommand{\zvec}{{\mathbf z}} \newcommand{\rvec}{{\mathbf r}} \newcommand{\mvec}{{\mathbf m}} \newcommand{\zerovec}{{\mathbf 0}} \newcommand{\onevec}{{\mathbf 1}} \newcommand{\real}{{\mathbb R}} \newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]} \newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]} \newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]} \newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]} \newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]} \newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]} \newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]} \newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]} \newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]} \newcommand{\laspan}[1]{\text{Span}\{#1\}} \newcommand{\bcal}{{\cal B}} \newcommand{\ccal}{{\cal C}} \newcommand{\scal}{{\cal S}} \newcommand{\wcal}{{\cal W}} \newcommand{\ecal}{{\cal E}} \newcommand{\coords}[2]{\left\{#1\right\}_{#2}} \newcommand{\gray}[1]{\color{gray}{#1}} \newcommand{\lgray}[1]{\color{lightgray}{#1}} \newcommand{\rank}{\operatorname{rank}} \newcommand{\row}{\text{Row}} \newcommand{\col}{\text{Col}} \renewcommand{\row}{\text{Row}} \newcommand{\nul}{\text{Nul}} \newcommand{\var}{\text{Var}} \newcommand{\corr}{\text{corr}} \newcommand{\len}[1]{\left|#1\right|} \newcommand{\bbar}{\overline{\bvec}} \newcommand{\bhat}{\widehat{\bvec}} \newcommand{\bperp}{\bvec^\perp} \newcommand{\xhat}{\widehat{\xvec}} \newcommand{\vhat}{\widehat{\vvec}} \newcommand{\uhat}{\widehat{\uvec}} \newcommand{\what}{\widehat{\wvec}} \newcommand{\Sighat}{\widehat{\Sigma}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
1
Vectors and Matrices
1.1
Scalar Multiplication
1.2
linear combinations
1.3
The dot product
1.3.1
The geometry of the dot product
1.4
Matrix vector multiplication
2
Matrix Multiplication
2.1
matrix-matrix products
2.1.1
Matrix-matrix products
2.2
Matrix transformations
2.3
Dynamical Systems
3
Matrix transformations
3.1
matrix transformations geometry
3.2
Rotations, reflections and shears
3.3
Bakery Application
4
Linear systems of equations
4.1
The equation
\(A\xvec = \bvec\)
4.2
The geometry of linear systems of equations in two and three unknowns
4.3
Gaussian Elimination
4.4
Augmented matrices
5
Reduced row echelon form
5.1
Reduced row echelon form
5.2
Using Sage to find RREF
5.3
Pivot positions, free and basic variables
5.4
Expressing solutions to the equation
\(A\xvec = \bvec\)
in terms of the vector
\(\xvec\)
6
Span
6.1
Existence of solutions and span
6.2
The span of vectors in
\(\real^2\)
6.3
The span of vectors in
\(\real^3\)
7
Linear independence
7.1
Uniqueness of solutions and linear independence
7.1.1
Linear dependence
7.2
How to recognize linear dependence
7.3
Linear independence and homogeneous equations
7.3.1
Homogeneous equations
7.3.2
Summary
8
Orthogonality and least squares
8.1
The dot product
8.1.1
The geometry of the dot product
8.2
The matrix transpose
8.3
Orthogonal bases
8.4
Orthogonal Projections
8.5
Orthogonal least squares
8.5.1
A first example
8.5.2
Solving least squares problems
9
Invertibility
9.1
Invertible matrices
9.2
Finding
\(A^{-1}\)
and using it to solve equations
9.3
Determinants: geometry and invertibility
9.3.1
Determinants of
\(2\times2\)
matrices
9.3.2
Determinants and invertibility
9.4
Cofactor expansions for computing determinants
10
Eigenvalues and Eigenvectors
10.1
Subpaces
10.1.1
Summary
10.2
Finding eigenvalues
10.3
Finding eigenvectors
10.4
Eigenvalues and their multiplicities
10.5
Diagonalization and powers of a matrix
10.5.1
Powers of a diagonalizable matrix
Backmatter
Colophon
Colophon
This book was authored in PreTeXt.
