\documentclass[11pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb,amscd}

\pagestyle{headings}
\setlength{\textwidth}{468true pt}
\setlength{\headheight}{0true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{0 truept}
\setlength{\evensidemargin}{0 truept}
%\setlength{\textheight}{580true pt}
\setlength{\textheight}{580true pt}

\begin{document}

\noindent
{\bf LINEAR ALGEBRA (MATH 3130)}

\noindent
{\bf SECTION 2, SPRING 2013}

\noindent
{\bf REVIEW SHEET}

\noindent
From the book: Sections 1.1--6.5, excluding Sections 4.8, 5.6--5.8.


\bigskip

\begin{enumerate}
\item[I.] Systems of linear equations.

\begin{enumerate}
\item[(a)] Augmented matrix and coefficient matrix of a system.
\item[(b)] Row reduction. (Reduced) row echelon form. Pivots,
pivot positions, pivot columns.
\item[(c)] Free and basic(=pivot) variables. Solutions sets.
Parametrized form of a solution.
\item[(d)] Consistent and inconsistent systems.
\item[(e)] Homogeneous systems. Relationship between 
solutions of $A{\bf x}={\bf b}$ and solutions of $A{\bf x}={\bf 0}$.
\end{enumerate}

\medskip

\item[II.] Matrix arithmetic.

\begin{enumerate}
\item[(a)] Matrices can be added, negated, multiplied with each other,
and scaled, provided the dimensions are right.
\item[(b)] The collection of $n\times n$ real matrices forms an
``$\mathbb R$-algebra'',
which is noncommutative if $n>1$. (This statement indicates
which laws of arithmetic are valid for $n\times n$ natrices, 
namely the ring laws. These laws are enumerated in Theorems 1 and 2
on pages 93 and 97.)
\item[(d)] Matrix transpose.
\item[(e)] Left and right inverses. (Equivalent properties.)
Two-sided inverses. A 1-sided invertible matrix
is 2-sided invertible iff it is square.
\item[(f)] Elementary matrices.
Row reduction is expressible as left multiplication
by a sequence of elementary matrices. Matrices 
$A$ and $B$ are row equivalent
iff $A=LB$ for some invertible matrix $L$.
\item[(g)] Algorithm for finding inverses.
\item[(h)] Partitioned matrices. Block diagonal and block 
triangular matrices.
\end{enumerate}

\medskip

\item[III.] Vectors and vector spaces.

\begin{enumerate}
\item[(a)] Linear systems may be viewed as vector equations.
\item[(b)] Definition of vector space. Definition of subspace.
\item[(c)] Geometric interpretation of vector space operations.
\item[(d)] Definition of column space and nullspace of a matrix.
\item[(e)] Spanning set of vectors.
\item[(f)] Linearly (in)dependent set of vectors.
\end{enumerate}

\medskip

\item[IV.] Linear Transformations.

\begin{enumerate}
\item[(a)] Definition. Fact that any linear transformation
has the form $T({\bf x})=A{\bf x}$.
\item[(b)] The problem of solving the linear system $A{\bf x}={\bf b}$ 
may be viewed as the problem of finding a vector
${\bf x}\in T^{-1}({\bf b})$ for $T({\bf x})=A{\bf x}$.
\item[(c)] One-to-one and onto transformations.
\item[(d)] Finding the standard matrix of a transformation.
\item[(e)] Matrices for rotation and reflection in the plane.
\end{enumerate}

\medskip

\item[V.] Matrix factorization.

\begin{enumerate}
\item[(a)] $LU$ factorization. 
\item[(b)] Solving systems with an $LU$ factorization.
\end{enumerate}

\medskip

\item[VI.] Applications.

\begin{enumerate}
\item[(a)] Balancing a chemical reaction.
\item[(b)] Network flow.
\item[(c)] Predator-prey dynamics.
\item[(d)] Leontief economic model.
\end{enumerate}

\medskip

\item[VII.] Affine transformations.

\begin{enumerate}
\item[(a)] An affine transformation has the form 
$T({\bf x}) = A{\bf x}+{\bf t}$. Examples: 
translations, plane reflections about an arbitrary line, 
and plane rotations about an arbitrary point.
\item[(b)] 
Affine transformations
in $\mathbb R^n$ may be represented in homogeneous coordinates
in $\mathbb R^{n+1}$ by matrices of the form 
${\tiny \left[\begin{array}{c|c}A&{\bf t}\\ \hline
{\bf 0}&1\\ \end{array}\right]}$.
\end{enumerate}

\medskip

\item[VIII.] Subspaces

\begin{enumerate}
\item[(a)] The structure of subspaces of $\mathbb R^n$.
\item[(b)] Four fundamental subspaces:
${\rm Col}(A)$, 
${\rm Nul}(A)$, 
${\rm Row}(A)={\rm Col}(A^T)$, and
${\rm Nul}(A^T)$.
\item[(c)] 
Ordered and unordered bases for a subspace.
\item[(d)] 
Algorithms for finding bases for
${\rm Nul}(A)$, 
${\rm Row}(A)$, and
${\rm Col}(A)$.
\item[(e)] 
Standard basis for $\mathbb R^n$.
\item[(f)] 
Dimension of a subspace.
\item[(g)] 
Rank and nullity of a matrix. Rank $+$ nullity theorem.
\item[(h)] 
Proof that ``dimension'' is well defined, namely, that
the size of any independent set is less or equal the size
of any spanning set, and that a maximal independent
set is spanning while a minimal spanning set is independent.
\end{enumerate}

\medskip

\item[IX.] The determinant.

\begin{enumerate}
\item[(a)] Signed volume.
\item[(b)] Minor, cofactor, 
definition of the determinant via the Laplace expansion.
\item[(c)] $\det(A)$ is defined only if $A$ is square.
$\det(A)\neq 0$ iff the columns of $A$ are independent. 
\item[(d)] 
Adjugate matrix. Fact that $A\cdot {\rm adj}(A) = \det(A)\cdot I$,
hence $A^{-1}=(1/\det(A)){\rm adj}(A)$ when $A$ is invertible.
\item[(e)] Further properties: 
$\det(AB)=\det(A)\det(B)$, the determinant can be computed
by Gaussian elimination, the determinant of a block triangular matrix
if the product of the determinants of the blocks, 
if $T({\bf x})=A{\bf x}$, 
then the determinant of $A$ measures the 
``volume expansion'' associated with $T$.
\item[(f)] 
``Correct'' definition:
the determinant is the unique alternating multilinear 
function $d$ of $n$ variables defined on $\mathbb R^n$ 
for which $d({\bf e}_1,\ldots,{\bf e}_n)=1$.
\item[(g)] 
Permutation expansion of the determinant.
Fact that $\det(A)=\det(A^T)$.
\item[(h)] Cramer's Rule for solving a linear system
$A{\bf x}={\bf b}$ with invertible $A$.
\end{enumerate}

\medskip

\item[X.] Abstract vector spaces.

\begin{enumerate}
\item[(a)] Meaning of the word ``abstract''.
\item[(b)] Definition and examples of abstract
vector spaces, e.g., $M_{m\times n}(\mathbb R)$,
$\mathbb P_n(t)$, $C^k([0,1])$. We
computed that 
$M_{m\times n}(\mathbb R)$,
has dimension $mn$, 
$\mathbb P_n(t)$ 
has dimension $n+1$, and 
that $C^0([0,1])$
must be infinite dimensional. 
\item[(c)] Coordinates relative to a basis.
\item[(d)] Definition of ``isomorphism'' of vector
spaces. Proof that every finitely generated real
vector space is isomorphic to $\mathbb R^n$ for some 
finite $n$.
\item[(e)] Matrices, ${}_{\mathcal C}[T]_{\mathcal B}$,
for linear transformations between 
abstract vector spaces. Change of basis matrices, 
${}_{\mathcal C}[I]_{\mathcal B}$.
\end{enumerate}

\medskip

\item[XI.] Markov chains.

\begin{enumerate}
\item[(a)] Definitions of: probability vector, 
(left, right, doubly) stochastic matrix and Markov chain.
\item[(b)] Steady state vector.
\item[(c)] Regular stochastic matrices have a unique 
steady state vector that is a probability vector.
Its entries are strictly positive.
\end{enumerate}

\medskip

\item[XII.] Eigenvalues, eigenvectors, eigenspaces.

\begin{enumerate}
\item[(a)] Eigenvectors identify ``preserved directions''
of a linear transformation $T\colon V\to V$.
\item[(b)] Definitions of eigenvector, eigenvalue, eigenspaces.
\item[(c)] Methods of calculation:
characteristic polynomial 
$\chi_A(\lambda)$ equals $\det(A-\lambda I)$; e-values of $A$
are the roots of $\chi_A(\lambda)=0$; 
e-space $V_{\lambda}$ equals ${\rm Nul}(A-\lambda I)$;
$\lambda$-eigenvectors are the nonzero vectors of 
$V_{\lambda}$. Fast calculation of e-values for (block)
triangular matrices.
\end{enumerate}

\medskip

\item[XIII.] Diagonalization.

\begin{enumerate}
\item[(a)] 
Structure of roots of a real polynomial 
over $\mathbb R$ or $\mathbb C$, and of a complex
polynomial over $\mathbb C$. Algebraic multiplicity
of an e-value.
\item[(b)] Geometric multiplicity of an e-value.
\item[(c)] Defn. of ``diagonalizable''.
Thm. A transformation $T\colon V\to V$ is diagonalizable
iff $V$ has a basis consisting of e-vectors for $T$ iff
the geometric multiplicity of each e-value equals its
algebraic multiplicity.
\item[(d)] Independence of subspaces. Sums 
of subspaces and direct sums 
of independent subspaces.
A sum of distinct e-spaces is direct.
$T\colon V\to V$ is diagonalizable iff 
$V = \bigoplus_{\lambda} V_{\lambda}$.
(Side observation: 
${\rm dim}(U\oplus W) = {\rm dim}(U) + {\rm dim}(W)$.)
\item[(e)] Similarity: $A$ is similar to $B$
if $A$ is a conjugate of $B$, i.e., $A=S^{-1}BS$.
Similarity is an equivalence relation
on the set of $n\times n$ matrices. Matrices are similar iff they 
represent the same transformation relative to different bases.
Similar matrices have the same characteristic polynomial,
hence same e-values.
If $A = S^{-1}BS$, then $S\colon V_{\lambda}^A\to V_{\lambda}^B$
is an isomorphism for each e-value $\lambda$. $A$
is diagonalizable iff it is similar to a diagonal matrix.
\item[(f)] (The nondiagonalizable case.)
Generalized e-spaces $V^{(\infty)}_{\lambda} = 
\bigcup_k {\rm Nul}(A-\lambda I)^k$. 
\item[(g)] The algebraic multiplicity of $\lambda$ 
is $\dim(V^{(\infty)}_{\lambda})$.
\item[(h)] If $T\colon V\to V$ is defined over $\mathbb C$
and $V$ is f.g., then
$V$ is the direct sum of its generalized e-spaces.
\item[(i)] Jordan block, Jordan canonical form.
Eigenchains in $V^{(\infty)}_{\lambda}$ yield JCF.
Every transformation defined over $\mathbb C$ is similar to a 
matrix in JCF, and the JCF is unique up to a permutation of Jordan blocks.
\item[(j)] 
The JCF of $A$ can be determined indirectly from the numbers
`${\rm nullity}(A-\lambda I)^k$' for all $\lambda$ and $k$.
\item[(k)] Diagonalization and JCF of $T$ over $\mathbb R$
instead of $\mathbb C$:
$\overline{V_{\lambda}^{(\infty)}}=V_{\overline{\lambda}}^{(\infty)}$,
and 
$V_{\lambda}^{(\infty)}\oplus V_{\overline{\lambda}}^{(\infty)}$
has a nice real basis consisting of the real and imaginary
parts of the vectors in the $\lambda$-eigenchains
in $V_{\lambda}^{(\infty)}$.
This choice of basis results in diagonal form or JCF
with $2\times 2$ real blocks replacing pairs of $1\times 1$
conjugate complex blocks.
\end{enumerate}

\medskip

\item[XIV.] Orthogonality.

\begin{enumerate}
\item[(a)] Dot product. (Defn. Arithmetic facts follow
from those of matrices, since
${\bf u}\bullet{\bf v}={\bf u}^T{\bf v}$.)
\item[(b)] Length in $\mathbb R^n$. Unit vector in direction ${\bf v}$
is ${\bf v}/{\|{\bf v}\|}$.
\item[(c)] Angle in $\mathbb R^n$ via 
${\bf u}\bullet{\bf v}=\|{\bf u}\| \|{\bf v}\| \cos(\theta)$.
Cauchy-Schwarz Inequality.
\item[(d)] Orthogonality. Orthogonal complement.
${\rm Row}(A)^{\perp} = {\rm Nul}(A)$. Algorithm for 
finding the orthogonal complement of a set of vectors.
\item[(e)] Orthonormal set of vectors. Angle-preserving linear
transformations. Orthogonal matrices.
\item[(f)] Approximate solutions to $A{\bf x}={\bf b}$ via least 
squares. Normal equations $A^TA{\bf x}=A^T{\bf b}$. 
Fitting curves to data.
\item[(g)] Orthogonal projection onto a vector or subspace.
Gram-Schmidt algorithm.
\end{enumerate}
\end{enumerate}

\bigskip

\noindent
{\bf General advice on preparing for a math test.} 

\noindent
Be prepared to 
demonstrate understanding in the following ways.
\begin{enumerate}
\item[(i)] 
Know the definitions of new concepts, and
the meanings of the definitions.
\item[(ii)] 
Know the statements and meanings
of the major theorems.
\item[(iii)] 
Know examples/counterexamples.
(The purpose of an example is to illustrate the 
extent of a definition or theorem.
The purpose of a counterexample is 
to indicate the limits of a definition or theorem.)
\item[(iv)] Know how to perform the different kinds of calculations
discussed in class.
\item[(v)] Be prepared to prove elementary statements.
(Understanding the proofs done in class is the best preparation
for this.)
\item[(vi)] Know how to correct mistakes made on old HW.
\end{enumerate}

\bigskip

\noindent
{\bf Sample Problems.} 
\begin{enumerate}
\item
Chapter 1--6 supplementary problems (excluding problems marked [{\bf M}]).

\medskip
\item
Let $A$ be a square matrix.
Explain why if the columns of $A$ 
are independent, then the columns of $A^2$ are independent.

\medskip
\item
Show that if $\{{\bf a}, {\bf b}, {\bf c}\}$ is
independent, then $\{{\bf a}, {\bf a}+{\bf b}, {\bf a}+{\bf b}+{\bf c}\}$
is also independent.

\medskip
\item
Explain why if $A$ and $B$ are $n\times n$ matrices
satisfying $A{\bf x}=B{\bf x}$ for all vectors
${\bf x}\in \mathbb R^n$, then $A=B$.


\medskip
\item
Use the definition of ``linear transformation'' to show
that the composition of two linear transformations
is a linear transformation.

\medskip
\item
Among all $n\times n$ matrices whose entries are 
all either $0$ or $1$ what is the maximum 
possible number of $1$'s
if the matrix is invertible? What is the minimum number
of $1$'s if the matrix is invertible?
For which values of $n$ is it possible for
the number of $1$'s to be equal to the number of $0$'s
and still have the matrix invertible?

\medskip
\item
Explain why if $S = \{{\bf v}_1,\ldots, {\bf v}_k\}$
is a subset of $\mathbb R^n$ that is linearly independent 
and spans the space, then $k=n$.

\medskip
\item
Which matrices commute with 
$\left[\begin{array}{cc}1&2\\3&4\end{array}\right]$?
(Hint: set up a linear system.)

\medskip
\item
Explain why the set of columns of an $n\times n$ invertible matrix
spans $\mathbb R^n$. Then explain why this set 
of columns is independent.

\medskip
\item
Explain why every $n\times n$ matrix $M$ is expressible
in exactly one way as a sum $M=S+A$ where $S$
is symmetric and $A$ is antisymmetric.

\medskip
\item
Show that if $A$ is invertible, then $A$
has at most one $LU$ factorization.

\medskip
\item
Give an example of a square matrix with no $LU$ factorization.

\medskip

\item
Computational problems:
\begin{enumerate}
\item
Using homogeneous coordinates,
find a matrix representation for the transformation
that rotates the plane $45^{\circ}$ around the point 
$\left[\begin{array}{r} 1\\2\end{array}\right]$.
\item
Find bases for the null space, row space and column space
of the $3\times 3$ matrix whose entries are all $1$.
What are the dimensions of these spaces?
\item
Put the numbers $1, 2, \ldots, 9$ into a $3\times 3$
matrix in order. What is the determinant?
\item
Find a change of basis matrix from 
${\mathcal B}=\left(
\left[\begin{array}{c} 1\\1\\0\end{array}\right],
\left[\begin{array}{c} 1\\0\\0\end{array}\right],
\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)$
to \\
${\mathcal C}=\left(
\left[\begin{array}{c} 0\\0\\1\end{array}\right],
\left[\begin{array}{c} 0\\1\\1\end{array}\right],
\left[\begin{array}{c} 1\\1\\1\end{array}\right]\right)$.
\item
Find the characteristic equation, e-values,
and e-spaces of $A=\left[\begin{array}{cc} 1&2\\2&4\end{array}\right]$.
Find a matrix $S$ that conjugates $A$ into diagonal form.
\item
Find a basis for $\left\{\left[\begin{array}{c} 1\\2\\3\end{array}\right],
\left[\begin{array}{c} 3\\2\\1\end{array}\right]
\right\}^{\perp}$.
\end{enumerate}

\medskip

\item
Can any of the following exist? (If so, give an example,
if not give a reason.)
\begin{enumerate}
\item
A vector space with an empty basis.
\item
A matrix of rank zero.
\item
A matrix with no determinant.
\item
A matrix with a zero dimensional eigenspace.
\item
An invertible matrix whose row sums are all zero.
\item
A real 
matrix whose null space equals its column space.
\item
A matrix $A$ such that ${\rm nullity}(A)=1$ and ${\rm nullity}(A^2)=3$.
\item
A matrix
where the dimension of the row
space is greater than the dimension
of the column space.
\item
A real number that does not arise as the determinant
of a real matrix.
\item
A vector space with no subspaces.
\item
An isomorphism between vector spaces of different dimensions.
\item
A matrix whose row space is isomorphic to its column space.
\item
A matrix whose characteristic polynomial is $\lambda^2+\lambda+1$.
\item
An eigenvalue whose geometric multiplicity exceeds
its algebraic multiplicity.
\item
A real $10\times 10$ matrix with 
only one eigenvector.
%No! Scalar multiples!
\item
A matrix equal to its adjugate.
\item
A left stochastic matrix that is not right stochastic.
\item
Conjugate matrices of different ranks.
\item
Conjugate matrices that are not similar.
\item
A matrix that is diagonalizable over $\mathbb C$
but not diagonalizable over $\mathbb R$.
\item
An orthogonal basis for $\mathbb R^3$ that is not orthonormal.
\item
A nondiagonalizable complex matrix.
\item
An orthogonal matrix with determinant zero.
\item
A $3\times 3$ orthogonal matrix with no zero entries.
\item
Subspaces $U$ and $W$ such that $U+W\neq U\oplus W$.
\item
A real vector that is orthogonal to itself.
\end{enumerate}

\medskip

\item
Give the dimensions of the following real vector spaces.
\begin{enumerate}
\item
The space of real polynomials $p(t)$ of degree at most $3$
which satisfy $p(1) = p(-1) = 0$.
\item
The space of $3\times 3$ upper triangular real matrices.
\item
The space of twice continuously differentiable functions $y=f(x)$
satisfying $y'' = 0$.
\end{enumerate}

\medskip
\item
How would you solve the following problem?
Suppose that $V$ has basis $({\bf v}_1,\ldots,{\bf v}_n)$
and $U$ is a subspace of $V$ with basis 
$({\bf u}_1,\ldots,{\bf u}_m)$. How do you find a basis
for $V$ whose first $m$ vectors form a basis for
$U$?

\medskip
\item
Suppose that you are given bases $\mathcal B$
and $\mathcal C$ for subspaces 
$U$ and $W$ of a space $V$. How would you find a basis
for $U+W$?  
How would you find a basis
for $U\cap W$? 
(Hint: in both cases, you should apply Gaussian
Elimination to the matrix $[{\mathcal B}|{\mathcal C}]$.
How should you use the results?)

\medskip
\item
Is there a $3\times 3$ matrix whose minors are nonzero and all equal?
Is there a $3\times 3$ matrix whose cofactors are nonzero and all equal?


\medskip
\item
Let $S$ be a $2\times 2$ invertible matrix.
Consider the linear transformation of ``conjugation by $S$'':
\[T\colon M_{2\times 2}(\mathbb R)\to M_{2\times 2}(\mathbb R)\colon
A\mapsto S^{-1}AS.\]
Show that if $\lambda$ is an e-value of $T$, then so is
$\lambda^k$ for any $k$. Show that 0 is not an e-value of $T$.
Explain why the e-values of $T$ can only be $+1$ or $-1$.
Show that $+1$ occurs as an e-value with multiplicity at least $2$.

\medskip
\item
What is the characteristic polynomial for the $n\times n$
matrix whose entries are all $1$?

\medskip
\item
Show that $\dim(U+W) = \dim(U)+\dim(W)-\dim(U\cap W)$.
(Solution 1 hint: choose a basis for $U\cap W$
and extend it in different ways to bases for both $U$ and $W$.
Show that all the vectors together form a basis for $U+W$.)
(Solution 2 hint: let $\mathcal B$
and $\mathcal C$ be bases for $U$ and $W$. 
Apply the rank+nullity theorem to the matrix $[{\mathcal B}|{\mathcal C}]$.)


\medskip
\item
The points 
$\left[\begin{array}{r} 0\\0\\0\end{array}\right]$,
$\left[\begin{array}{r} 1\\1\\0\end{array}\right]$,
$\left[\begin{array}{r} 1\\0\\1\end{array}\right]$, and
$\left[\begin{array}{r} 0\\1\\1\end{array}\right]$ are the vertices
of a regular tetrahedron. Find the lengths of the sides
and the angles formed by adjacent faces.

\medskip
\item
Find the least squares curve of the form 
$y=ax^2+bx+c$ that best fits the data points\\
$(-2,2), (-1,1), (0,0), (1,1), (2,2)$.

\medskip
\item
Show that if $V$ is finite dimensional and $U$ is a subspace,
then $V = U\oplus U^{\perp}$.

\medskip
\item
Show that $(U+W)^{\perp}=U^{\perp}\cap W^{\perp}$.

\medskip
\item
Use the Gram-Schmidt process to find an orthonormal 
basis for the subspace of $\mathbb R^3$ spanned by 
$\left[\begin{array}{r} 0\\4\\2\end{array}\right]$
and
$\left[\begin{array}{r} 5\\6\\-7\end{array}\right]$.

\end{enumerate}
\end{document}
