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\begin{document}
\examtitle{MATH 2270-4}{Exam 3}{S2019}
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\studentinfo
\instructions{Please, no books, notes or electronic devices.\\[1pc]
Some questions involve proofs. Please divide your time accordingly.\\[1pc]
Extra details can appear on the back side or on extra pages. Please supply a road map for details not directly following the problem statement.\\[1pc]
 Details count 75\% and answers count 25\%.
\\
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{\bf Definition}: An {\em abstract} vector space $V$ is a data set of packages called {\bf vectors} together with operations of addition ($+$) and scalar multiplication ($\cdot$) satisfying the following eight (8) rules:

\begin{minipage}{\textwidth}\sf
 {\bf Closure}: If $\.x$ and $\.y$ are in $V$, then $\.x+\.y$ is defined and in $V$.\\
 {\bf (1)}~ $\.x+\.y=\.y+\.x$\\
 {\bf (2)}~ $\.x+(\.y+\.z)=(\.x+\.y)+\.z$\\
 {\bf (3)}~ There is a zero vector $\.0$ in $V$ with $\.x+\.0=\.x$.\\
 {\bf (4)}~ There is a vector $-\.x$ in $V$ with $\.x+(-\.x)=\.0$.\\
 {\bf Closure}: If $c$ = constant and $\.x$ is in $V$, then $c \cdot \.x$ is defined and in $V$.\\
 {\bf (5)}~ $a(\.x+\.y)=a\.x+a\.y$ \\
 {\bf (6)}~ $(a+b)\cdot \.x = a\cdot \.x + b\cdot \.x$\\
 {\bf (7)}~  $(ab)\cdot \.x = a \cdot (b\cdot \.x)$\\
 {\bf (8)}~ $1\cdot \.x = \.x$
\end{minipage}


\begin{minipage}{\textwidth}\small\sf
{\bf Definition}. If vectors $\.b_1,\.b_2,\ldots,\.b_n$ are a basis for subspace $X$ of an {\em abstract} vector space $V$, and $\.x=c_1\.{b}_1 + c_2\.{b}_2+ \cdots +c_n\.{b}_n$ is a given linear combination of these vectors, then the uniquely determined constants $c_1,c_2,\ldots,c_n$ are called the {\em coordinates of $\.{x}$ relative to the
basis} $\.{b}_1, \.{b}_2, \ldots,\.{b}_n$ and the {\em coordinate map} is the isomorphism
$$\.x=c_1\.{b}_1 + c_2\.{b}_2+ \cdots+c_n\.{b}_n \to \colvectorC{c_1\\ c_2 \\ \vdots \\ c_n}.$$
\end{minipage}

\medskip

{\bf Definition}: A subset $S$ of a vector space $V$ is a {\bf subspace} of $V$ provided\\[1ex]
\begin{minipage}{\textwidth}\sf
 {\bf (1)}~ The zero vector is in $S$ \\
 {\bf (2)}~ If vectors $\.x$ and $\.y$ are in $S$, then $\.x+\.y$ is in $S$. \\
 {\bf (3)}~ If vector $\.x$ is in $S$ and $c$ is any scalar, then $c\.x$ is in $S$.
\end{minipage}


\medskip

{\bf Definition}: Vectors $\.v_1,\ldots,\.v_p$ in an abstract vector space $V$ are said to be {\bf independent} in $V$ provided solving the equation $c_1\.v_1+\cdots+c_p\.v_p=\.0$ for scalars $c_1,\ldots,c_p$ has only the zero solution
$c_1=\cdots=c_p=0$.


\anotherpage

\problem{100}{
Let $V$ be the vector space of all functions on $(-\infty,\infty)$.
Define $W=\SPAN\{x,e^x\}$. Assume known that $x,e^x$ are independent functions.
Define subspace $S=\SPAN\{\.{v}_1,\.{v}_2\}$ where
$$\.{v}_1:~ y=x+e^x,\quad \.{v}_2:~ y=x-e^x.$$
{\bf (a)} [20\%] Explain why $S$ is contained in $W$, that is, provide details
for why linear combinations of vectors $\.{v}_1,\.{v}_2$ are in $W$.\\
\rule{0.9\textwidth}{0.3ex}

\myspace{10pc}

{\bf (b)} [40\%] Prove that $W=S$. Therefore $\dim(S)=\dim(W)=2$, which proves independence of vectors
$\.{v}_1:~ y=x+e^x,\quad \.{v}_2:~ y=x-e^x.$\\
\rule{\textwidth}{0.3ex}

\anotherpage

{\bf (c)} [40\%] Define vector $\.v$ in $S$ by equation $y=2x+3e^x$.  {\bf Show how } to compute $d_1$, $d_2$ in the equation $\.{v}=d_1\.v_1+d_2\.v_2$, {\bf using coordinate map methods}. The definitions are $\.{v}_1:~ y=x+e^x,\quad \.{v}_2:~ y=x-e^x.$\\

{\bf Expected in (c)}: {\small\sf Calculations of $d_1,d_2$ are to be done using column vectors from $\Rr^2$ and $2\times 2$ matrices, not functions
from $V$}. {\bf Zero credit} for not using column vectors and coordinate maps.\\
\rule{\textwidth}{0.3ex}
}{}{~}
%A:=<1,1,0|2,0,1|2,1,1>; v:=<3,4,1>; LinearSolve(A,v); c1,c2,c3=[1,-2,3];
%
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\begin{small}
{\bf Definition}: A subset $S$ of a vector space $V$ is a {\bf subspace} of $V$ provided\\[1ex]
\begin{minipage}{\textwidth}\sf
 {\bf (1)}~ The zero vector is in $S$ \\
 {\bf (2)}~ If vectors $\.x$ and $\.y$ are in $S$, then $\.x+\.y$ is in $S$. \\
 {\bf (3)}~ If vector $\.x$ is in $S$ and $c$ is any scalar, then $c\.x$ is in $S$.
\end{minipage}
\end{small}

\problem{100}{~\newline
{\bf (a)} [60\%] Let $V$ be an {\em abstract} vector space.
           Let $\.v_1,\.v_2$ be two vectors in $V$. Define $S$ to be the set of all linear combinations of $\.v_1 + \.v_2, \.v_1-\.v_2$. Prove that
           $S$ is a subspace of $V$, using only the definition of subspace.
\begin{quote}
{\bf Expected}: {\small\sf A proof uses the symbols $\.v_1,\.v_2$ and the 8 rules of a vector space, plus theorems like $0\.v=\.0$. Symbols $\.v_1,\.v_2$ are not assumed to be column vectors.}\\
\rule{0.9\textwidth}{0.3ex}
\end{quote}

\relax
\anotherpage
\begin{small}
{\bf Definition}: A subset $S$ of a vector space $V$ is a {\bf subspace} of $V$ provided\\[1ex]
\begin{minipage}{\textwidth}\sf
 {\bf (1)}~ The zero vector is in $S$ \\
 {\bf (2)}~ If vectors $\.x$ and $\.y$ are in $S$, then $\.x+\.y$ is in $S$. \\
 {\bf (3)}~ If vector $\.x$ is in $S$ and $c$ is any scalar, then $c\.x$ is in $S$.
\end{minipage}
\end{small}

{\bf (b)} [40\%] Let $V$ be the vector space of all $2\times 2$ matrices. Invent an example of a non-void subset $S$ of $V$ that satisfies (1) and (2) but fails the third item (3).\\
\rule{\textwidth}{0.3ex}
}{}{~}
%(1) If the zero vector is not in $S$, then $S$ is not a subspace.\\
%(2) If two vectors in $S$ fail to have their sum in $S$, then $S$ is not a subspace.\\
%(3) If a vector is in $S$ but its negative is not, then $S$ is not a subspace.
%
\anotherpage

\problem{100}{
Let $A$ be a $4\times 3$ matrix. Assume the determinant of $A^TA$ is zero. Prove  that the nullspace of $A$ contains a nonzero vector.\\
\rule{\textwidth}{0.3ex}
}{}{~}
\anotherpage
\problem{100}{~\\
{\bf (a)} [40\%] Define $\.{y}=\minicolvector{1\\ -1 \\ 1}$, $\.{u}=\minicolvector{1\\ 1\\ 0}$. Find the orthogonal projection vector $\.{v}$ (the shadow projection vector) of $\.{y}$ onto the direction of $\.{u}$.\\
\rule{\textwidth}{0.3ex}

\myspace{10pc}

{\bf (b)} [60\%] Let $\.v_1=\minicolvector{1\\ 1\\ 0}$, $\.v_2=\minicolvector{1\\ -1\\ 1}$, $\.x=\minicolvector{1\\ 1\\ 2}$. Check that $\.v_1,\.v_2$ are orthogonal and then compute $\.v=$ the vector projection of
 $\.x$ onto the subspace $S=\SPAN\{\.v_1,\.v_2\}$.


{\bf Reminder}: {\small\sf $\.v$ is the sum of two shadow projections.\\
\rule{\textwidth}{0.3ex}
 %(1) $\.x$ onto $\.v_1$ and (2) $\.x$ onto $\.v_2$.
 }

\myspace{10pc}

~
}{}{Use the formula $\.{v}=d\.{u}$ where $d=\dd\frac{\.{y}\cdot\.{u}}{\.{u}\cdot\.{u}}$.}
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\problem{100}{
Let $A$ be an $m\times n$ matrix and $\.b$ an $m\times 1$ vector.
 Let $W$ be the column space of
 $A$. Linear equations $A^TA\.z=A^T\.b$ are the {\bf normal equations} for the problem $A\.x=\.b$.

\medskip


{\bf (a)} [30\%] Let $A=\left(\begin{array}{cc}1&1\\1&1\\1&0\end{array}\right)$, $\.b=\colvector{1\\ -1\\ -1}$.\\ Display toolkit steps that verify $A\.x=\.b$ has no solution.\\
\rule{\textwidth}{0.3ex}

\myspace{12pc}

{\bf (b)} [30\%] Let $\.c=A\.z$ where $\.z$ is the unique theoretical solution of the normal equations. Explain with a figure: $\.c$ is the nearest point to $\.b$ in the column space $W$.\\
\rule{\textwidth}{0.3ex}

\anotherpage

{\bf (c)} [40\%] Let $A=\left(\begin{array}{cc}1&1\\1&1\\1&0\end{array}\right)$, $\.b=\colvector{1\\ -1\\ -1}$. Find vector $\.z$ in the normal equations.\\
\rule{\textwidth}{0.3ex}
}{}{~}
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\problem{100}{
The {\bf Fundamental Theorem of Linear Algebra} contains this statement: {\large\sf The row space and the null space of a matrix are orthogonal}. This means that $\.R \cdot \.N = 0$ for each vector $\.R$ in the row space and each vector $\.N$ in the null space.\\[1pc]
The {\bf four fundamental subspaces} in the Fundamental Theorem of Linear algebra are: (1) Nullspace of $A$, (2) Column Space of $A$, (3) Row space of $A$, 4) Nullspace of $A^T$.



{\bf (a)} [30\%] Define precisely the four fundamental subspaces. For example, the Nullspace of $A$ is the set of all solutions $\.x$ to the matrix equation $A\.x=\.0$.\\
\rule{\textwidth}{0.3ex}

\myspace{16pc}

{\bf (b)} [30\%] Assume $A$ is $20\times 12$ and has rank $10$. Equivalently, matrix $A$ has $10$ pivots . Report the dimensions
of the four fundamental subspaces.\\
\rule{\textwidth}{0.3ex}
\anotherpage

{\bf (c)} [40\%]  Let $A$ be an $m\times n$ matrix.  Let $\.C$ be a linear combination of the columns of $A$ and let $\.Y$ belong to the nullspace of $A^T$. Prove that $\.C\cdot\.Y=0$,
that is, $\.C$ and $\.Y$ are orthogonal.\\
\rule{\textwidth}{0.3ex}
}{}{~}
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