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\begin{document}
\examtitle{MATH 2270-2}{Final Exam}{Spring 2018}

\studentinfo


\instructions{No books, notes or electronic devices, please.\\
The questions have credits which reflect the time required to write the solution.\\
If you must write a solution out of order or on the back side, then supply a road map.\\
Solutions are expected to include readable and convincing details. A correct answer without details earns 25\%.\\
Expect about 3 to 10 minutes per problem. Final exam problems may have multiple parts.
}
\finishfirstpage
\anotherpage

%\section*{Chapter 1}


\problem{100}{Solve Linear Algebraic Equations.

Solve the system $A\.{u}=\.{b}$  defined by
$$
\left\{\begin{array}{lcl}
 2x_1+x_2+8x_3+x_4 + 2x_5 &=& 4\\
   x_1+3x_2+4x_3+x_4 + x_5 &=& 2\\
 2x_1+2x_2+8x_3+x_4 + x_5 &=& 4
\end{array}\right.
\quad\quad
\.{u}=\minicolvectorC{x_1\\[-1ex] x_2\\[-1ex] x_3\\[-1ex] x_4\\[-1ex] x_5}, \quad \.b=\minicolvectorC{4\\ 2\\ 4}.
$$
%A:=<2,1,8,1,2|1,3,4,1,1|2,2,8,1,1>^+;b:=<4,2,4>;with(LinearAlgebra):LinearSolve(A,b);ReducedRowEchelonForm(<A|b>);
%# rref(a)=[[1, 0, 4, 0, -1, 2], [0, 1, 0, 0, -1, 0], [0, 0, 0, 1, 5, 0]]
\begin{minipage}{\textwidth}\small\sf {\bf Expected}: (a) Augmented matrix and Toolkit steps to the reduced row echelon form (RREF). (b) Translation of RREF to scalar equations. Identify lead and free variables.  (c) Scalar general solution for $x_1$ to $x_5$. (d) Write vector formulas for the homogeneous solution $\.u_h$, a particular solution $\.u_p$ and the general solution $\.u=\.u_h+\.u_p$.\end{minipage}
\iffalse
\begin{enumerate}
\item[{\bf (a)}] [20\%] Find the augmented matrix $C$ and display the toolkit steps to the reduced row echelon form of $C$.
\end{enumerate}
\hrulefill

\myvspace{4in}
\begin{enumerate}
\item[{\bf (b)}] [20\%] Write the {\bf scalar equations} corresponding to $\rref(C)$. Then
identify the {\bf free} variables and the {\bf lead} variables.

\hrulefill
\myvspace{1.2in}
\item[{\bf (c)}] [20\%] Display the {\bf scalar} general solution and then the {\bf vector} general solution $\.{u}$.

\hrulefill
\myvspace{1.5in}

\item[{\bf (d)}] [20\%] Extract from the answer in (c) the {\bf vector formulas} for a particular solution $\.{u}_p$ and the homogeneous solution $\.{u}_h$.

 \hrulefill
\myvspace{1.5in}

\item[{\bf (e)}] [20\%] Extract from the answer in (d) a vector solution basis for the homogeneous problem $A\vec{u}=\.{0}$. These vectors
 are called {\bf Strang's Special Solutions}.

 \hrulefill

\end{enumerate}
\fi
}{}{}
{Chapter 1}


\anotherpage
\problem{100}{Inverse, Determinant, Cramer's Rule.

\begin{enumerate}
 \item[{\bf (a)}] [40\%] Find the inverse of the matrix $A=\threebythree{1&4&0}{1&5&0}{0&0&1}$.

 
 \item[{\bf (b)}] [30\%] Let $A$ be defined as in part (a). Compute the determinant of $((A+A^T)^{-1})^T$.


\item[{\bf (c)}] [30\%]
Let $P,Q,R$ be undisclosed real numbers. Define matrix $B$ and vectors $\.x$ and $\.{b}$ by the equations
\myvspace{-4ex}
$$B=\left(\begin{array}{rrrr}
 -2 &  0 & 0  \\
  0 & -1 &  1  \\
  2 &  1 &  2
 \end{array}\right), \quad \.x=\colvector{x_1\\ x_2\\ x_3}, \quad \.{b}=\colvector{P\\ Q \\ R }.$$
 \myvspace{-2ex}
Find the value of $x_3$ by Cramer's Rule
in the system $B\.{x}=\.{b}$.
% B:=<-2,0,0|0,-1,1|2,1,2>^+;Determinant(B);LinearSolve(A,<p,q,r>);

\hrulefill
 \end{enumerate}
}{}{in progress
}
{Chapters 2 and 3}
% A:=<1,4|1,5>^+; 1/A;


\anotherpage



\problem{100}{Independence Tests for Functions.
\vspace*{-2ex}
\begin{quote}\small\sf
{\bf Theorem (Wronskian test)}. Wronskian determinant of ${f}_1,{f}_2,{f}_3$ nonzero at some invented $x=x_0$ implies independence of ${f}_1,{f}_2,{f}_3$.

 {\bf Theorem (Sampling test)}. Functions ${f}_1,{f}_2,{f}_3$ are independent if a sampling matrix constructed for some invented samples
 $x_1,x_2,x_3$ has nonzero determinant.

\end{quote}
\vspace*{-2ex}
Let $V$ be the vector space of all functions on $(-\infty,\infty)$. Define functions in $V$ by the equations
$f_1(x)=5+e^x$, $f_2(x)=2x$, $f_3(x)=x + x^2$.


  {\bf (a)} [50\%] Construct the Wronskian matrix $W$ of the given functions $f_1,f_2,f_3$, then invent a value for $x$ such that $|W|\ne 0$.


 {\bf (b)} [50\%] Construct a sampling matrix $S$ for the given functions $f_1,f_2,f_3$ such that $|S|\ne 0$.

\hrulefill

}{}{in progress
}
{Chapters 1 to 4}

\anotherpage

\problem{100}{Independence Tests for Column Vectors.
\vspace*{-4ex}
\begin{center}\small\sf\begin{tabular}{lp{0.8\textwidth}}
\bf Rank test & Vectors $\.{v}_1,\.{v}_2,\.{v}_3$ are independent if their augmented matrix has rank 3. \\
\bf Determinant test & Vectors $\.{v}_1,\.{v}_2,\.{v}_3$ are independent if their square augmented matrix has nonzero determinant. \\
 \bf Pivot test & Vectors $\.{v}_1,\.{v}_2,\.{v}_3$ are independent if their augmented matrix $A$ has 3 pivot columns.\\[1ex]
\bf Orthogonality test & A set of nonzero pairwise orthogonal vectors is independent.\\[1ex]
 % \bf Combination test & A list of vectors is independent if each vector is not a linear combination of the preceding vectors.
\end{tabular}\end{center}
\vspace*{-2ex}
Let $V$ be the vector space of all functions on $(-\infty,\infty)$.
It is known that functions
$5+e^x$, $2x-e^x$, $x + e^x$
are independent in $V$. Let $S=\SPAN(5+e^x, 2x-e^x, x + e^x)$.
Define a coordinate map isomorphism from $S$ to $\Rr^3$ by
\myvspace{-4ex}
$$
T ~:~ c_1(5+e^x) + c_2(2x-e^x)+c_3(x + e^x)
\quad\mbox{maps into}\quad \minicolvector{c_1\\ c_2\\ c_3}.$$

\myvspace{-3ex}

{\bf (a)} [60\%]  Define functions in $V$ by the equations
$f_1(x)=x+e^x$, $f_2(x)=2x-e^x$, $f_3(x)=3 +  x + e^x$. Calculate the column vectors $\.{v}_1=T(f_1), \.{v}_2=T(f_2), \.{v}_3=T(f_3)$.

{\bf (b)} [40\%]
Because $T$ is one-to-one and onto, then the given functions $f_1,f_2,f_3$ are independent in $S$ if and only if the column vectors $T(f_1),T(f_2),T(f_3)$ are independent in $\Rr^3$.
Show details for {\bf one} of the independence tests applied to the column vectors $\.{v}_1, \.{v}_2, \.{v}_3$ calculated in part (a) above.

\hrulefill

}{}{in progress
}
{Chapters 1 to 4}
\anotherpage

\problem{100}{Subspace Definition and Theorems.
\vspace*{-3ex}
\begin{quote}\small\sf
{\bf DEFINITION}. Subset $S$ of vector space $V$ is a subspace of $V$ provided (1), (2), (3) hold:
\\
(1) $S$ contains vector $\.0$.\\
(2) If $\.x$ and $\.y$ are in $S$, then $\.x+\.y$ is  in $S$.\\
(3) If $c$ is a constant and $\.x$ is in $S$, then $c\.x$ is in $S$.
\end{quote}
\vspace*{-2ex}
{\bf (a)} [50\%] Set $S$ consists of all vectors
$\.{x}$  with components $x_1,x_2,x_3$ in vector space $\Rr^{3}$  such that $x_1+2x_3=x_2$ and $x_1+x_3=0$.
Supply proof details which verify that
$S$ is a subspace of $\Rr^3$. Cite a theorem or else verify conditions (1), (2), (3) in the DEFINITION.


{\bf (b)} [50\%] Set $S$ consists of all vectors $\.x$ in vector space $\Rr^4$ which are linear combinations of the vectors\vspace*{-3ex}
$$\.{v}_1=\minicolvector{1\\1\\1\\7},\quad
\.{v}_2=\minicolvector{1\\-1\\0\\1}, \quad
\.{v}_3=\minicolvector{2\\0\\1\\8}.$$
Supply proof details which verify that
$S$ is a subspace of $\Rr^4$. Cite a theorem or else verify conditions (1), (2), (3) in the DEFINITION.

\hrulefill

}{}{in progress
}
{Chapters 2, 4 and 6}
\anotherpage


%\section*{Chapter 4}

% chapter 4
\problem{100}{Gram-Schmidt Orthogonalization Process.

Let $S$ be the subspace of $\Rr^4$ spanned by the independent vectors
$$\.{x}_1=\left(\begin{array}{r}1\\ 1\\1\\0\end{array}\right),\quad
\.{x}_2=\left(\begin{array}{r} -1\\ 1\\0\\1\end{array}\right),\quad \.{x}_3=\left(\begin{array}{r}1\\1\\0\\1\end{array}\right).$$
Find a Gram-Schmidt orthogonal basis $\.{v}_1, \.{v}_2, \.{v}_3$
for subspace $S$.

\hrulefill
}{}{in progress
}
{Chapter 6}


\anotherpage

\problem{100}{Symmetric Matrices and the Invertible Matrix Theorem.

% Edited 27 apr
Let $A$ be an $m\times n$ matrix and assume that $A^TA$ is invertible. Prove that the
columns of $A$ are linearly independent.

\begin{minipage}{\textwidth}\small\sf {\bf Expected}: A referenced result from {\em "The Invertible Matrix Theorem"} should appear as a completely stated LEMMA, the proof of the LEMMA deferred to the textbook.\end{minipage}

\hrulefill

}{}{in progress}
{Chapters 1 to 6}


\anotherpage

%\section*{Chapter 6}

\problem{100}{Eigenanalysis and Diagonalization.

The matrix $A$ below has eigenvalues $2$, $8$ and $8$.
$$
 A=\left(\begin{array}{rrrr}
 12 & 4 & -1 \\
 -4 & 4 & -2 \\
 0 & 0 &  2
 \end{array}\right)
 $$
%A:=Matrix([[12 , 4 , -1 ],
%  [-4 , 4 , -2 ],
%  [0 , 0 ,  2  ]]);Eigenvalues(A);Eigenvectors(A);
{\bf (a)} [80\%] Compute all eigenpairs of $A$.

\begin{minipage}{\textwidth}\small\sf {\bf Expected}:
 For each eigenvalue $\lambda$, (1) Compute the RREF of $A-\lambda I$, (2) Compute all eigenvectors for $\lambda$ as Strang's solutions. \end{minipage}

{\bf (b)} [20\%] Is $A$ diagonalizable? Explain why or why not.

 \hrulefill

}{}{in progress
}
{Chapter 5}

\anotherpage


\problem{100}{Near Point Theorem and Least Squares.

 {\bf (a)} [60\%] Define $A=\left(\begin{array}{rr} -1&1\\[-1ex] 0&1\\[-1ex] 1&1\end{array}\right)$ and $\.{\bf b}=\colvector{3\\[-1ex]  1\\[-1ex]  2}$. Write the normal equations for the inconsistent problem $A\.{x}=\.{b}$ and solve for the least squares solution ${\hat{{\bf x}}}$.

  {\bf (b)} [20\%] Continue part (a). Compute  vector ${\hat{\bf b}}=A{\hat{{\bf x}}}$, which is the near point to $\.{\bf b}$ in the column space of $A$.

{\bf (c)} [20\%]
Least squares can be used to find the best fit line $y=mx+b$ for the $(x,y)$-data
points  $(-1, 3)$, $(0,1)$, $(1,2)$. Find the line equation by the method of least squares.


\begin{minipage}{\textwidth}\small\sf
{\bf Expected}: The matrix $A$ you create for part (c) should match the matrix $A$ of part (a). Save time by using the
computations from (a) and (b).
\end{minipage}

 \hrulefill

}{}{in progress
}
{Chapter 6}

% A:=<1,1|1,-1|1,0>^+;b:=<2,3,1>;LinearSolve(A,b);LinearSolve(A^+ . A,A^+ . b);LeastSquares(A,b);


\anotherpage


\problem{100}{Spectral Theorem and $AQ=QD$.

The spectral theorem says that a symmetric matrix $A$ satisfies $AQ=QD$ where $Q$ is orthogonal and $D$ is diagonal. Find $Q$ and $D$ for the symmetric matrix
$A=\begin{pmatrix} 7 & 3\\ 3 &7\end{pmatrix}$.
%with(LinearAlgebra):
% A:=<7,3|3,7>;Eigenvalues(A);Eigenvectors(A);
%#10,4
%Q:=(1/sqrt(2))*<1,1|-1,1>;DD:=<10,0|0,4>;A.Q-Q.DD;

\hrulefill
}{}{in progress\\

Start with the equation $r^2-10r+24=0$ having roots $r=4,6$. Compute the eigenpairs  $(4,\.{v}_1)$, $(6,\.{v}_2)$ where
$\.{v}_1=\colvector{1\\ 1}$ and $\.{v}_2=\colvector{-1\\ 1}$. The two vectors are orthogonal but not of unit length. Unitize them to
get $\.{u}_1=\frac{1}{\sqrt2}\.{v}_1$,  $\.{u}_2=\frac{1}{\sqrt2}\.{v}_2$. Then $Q=<\.{u}_1 | \.{u}_2>=\frac{1}{\sqrt2}\twobytwo{1&-1}{1&1}$,
$D=\diag(4,6)$.
}
{Chapter 7}

\anotherpage



\problem{100}{Singular Value Decomposition.

Determine the singular value decomposition for the matrix
$A=\twobytwo{6&2}{2&6}$.

\hrulefill
}{}{in progress
}
{Chapter 7}
%with(LinearAlgebra):
%A:=<6,2|2,6>;Eigenvalues(A);Eigenvectors(A);Determinant(A-r*IdentityMatrix(2));
%s,V1:=Eigenvectors(A^+ . A);sig:=<8,0|0,4>;
%V:=simplify(V1.(1/sqrt(2)));v1:=V[1..2,1];v2:=V[1..2,2];A.v1/8;A.v2/4;V^+ . V;
%u1:=A.v1/8;u2:=A.v2/4;U:=<u1|u2>;U.sig.V^+;
%#U, Vt := SingularValues(A, output = ['U', 'Vt']);
%evalf(U);evalf(V);

\anotherpage

\problem{100}{Linear Transformations as Matrix Multiply.

Let the linear transformation $T$ from $\Rr^2$ to $\Rr^2$
be defined by its action on two independent vectors:
$$
T\left(\begin{pmatrix}2\\ 3\end{pmatrix}\right)=
\begin{pmatrix}4\\ 3\end{pmatrix},\quad
T\left(\begin{pmatrix}1\\ 2\end{pmatrix}\right)=
\begin{pmatrix}4 \\ 1\end{pmatrix}
.$$
Find the unique $2\times 2$ matrix $A$ such that $T$ is defined
by the matrix multiply equation $T(\.{x})=A\.{x}$

\hrulefill
}{}{in progress\\
}
{Chapter 4}

\anotherpage
\problem{100}{Orthogonality.

Let $A=\left(\begin{array}{ccc} a & b & c\\[-1ex] d & e & f\end{array}\right)$ be a $2\times 3$ matrix, where $a,b,c,d,e,f$ are symbols that represent real numbers.
Define subspaces
\myvspace{-2ex}
$$\begin{array}{l}S_1 = \mbox{the column space of the transpose matrix $A^T$} = {\bf Col}(A^T)\\
    S_2 = \mbox{the null space of $A$} = {\bf Null}(A).\end{array}$$
Let $\.x$ belong to $S_1$ and let $\.y$ belong to $S_2$. Prove that their dot product is zero: $\.x \cdot \.y=0$.

\begin{minipage}{\textwidth}\small\sf
{\bf Expected}: Apply the definition of matrix multiply in terms of dot products. No theorems are used, only definitions.
\end{minipage}

\hrulefill
}{}{in progress
}
{Chapters 4 and 6}

\anotherpage
\problem{100}{Fundamental Theorem of Linear Algebra.

{\bf (a)} [40\%] Define precisely the four fundamental subspaces appearing in the Fundamental Theorem of Linear Algebra.

{\bf (b)} [30\%] What are the dimensions of the four subspaces?

{\bf (c)} [30\%]
 State the orthogonality relations for the four fundamental subspaces.

\hrulefill
}{}{Part 1. The dimensions are $n-r,r,rm-r$ for nullspace(A), colspace(A), rowspace(A), nullspace($A^T$).
Part 2. The orthogonality relation is rowspace $\perp$ nullspace, for both $A$ and $A^T$.
A full statement is expected, not the brief one given here.}
{Chapters 1 to 7}

\iffalse
\anotherpage
\problem{20}{
State the Fundamental Theorem of Linear Algebra. Include {\bf Part 1}:
The dimensions of the four subspaces, and {\bf Part 2}: The
orthogonality equations for the four subspaces.

\hrulefill
}{}{Part 1. The dimensions are $n-r,r,rm-r$ for nullspace(A), colspace(A), rowspace(A), nullspace($A^T$).
Part 2. The orthogonality relation is rowspace $\perp$ nullspace, for both $A$ and $A^T$.
A full statement is expected, not the brief one given here.}
{Chapters 1 to 7}


\anotherpage

\problem{20}{
State the {\bf Spectral Theorem} for symmetric matrices. Then write a paragraph about the important results developed for the spectral theorem: real eigenvalues, orthogonality, diagonalizability and the use of Gram-Schmidt.

\hrulefill
}{}{
A real symmetric $n\times n$ matrix $A$ has only real eigenvalues. Matrix $A$ has $n$ eigenpairs, in short it is is diagonalizable. To each eigenvalue of multiplicity $k$, there are $k$ independent eigenvectors. These eigenvectors span a subspace of dimension $k$ which by Gram-Schmidt is spanned by $k$ orthonormal vectors. Two such subspaces corresponding to different eigenvalues are orthogonal. The spectral decomposition of $A$ is $A=QDQ^T$ where $D$ is a diagonal matrix of eigenvalues and $Q$ is an orthogonal matrix of corresponding eigenvectors.
}
{Chapter 7}


\fi

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