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\begin{document}
\examtitle{MATH 2270-4}{Sample 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\%. Exam 3 on April 12  will have 6 questions, obtained from this sample by deleting two questions and editing the content below. Expect the April 12 exam to be like this sample, problems edited, but no change in background required.
\\
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% Problems Start Here % ----------------------------------------------------- %
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\problem{100}{%2018
Find the complete vector solution $\vec{x} = \vec{x}_h + \vec{x}_p$ for the nonhomogeneous system
$$\left(\begin{array}{rrrrrr}
0&3&1&0&0\\
0&3&3&0&6\\
0&0&0&0&0\end{array}\right)
\colvector{x_1 \\ x_2 \\ x_3\\ x_4 \\ x_5} = \colvector{3 \\ 1 \\ 0}.$$
{\bf Expected}:
{\small\sf (a) Augmented matrix. (b) Toolkit steps to RREF. (c) Translation of RREF to scalar equations. (d) Scalar general solution. (e) Find the homogeneous solution $\vec{x}_h$, which is a linear combination of Strang's special solutions. (f) Find a particular solution $\vec{x}_p$. (g) Write the vector general solution $\.x=\.x_h+\.x_p$.}
}{}{
}
\anotherpage
~\\
{\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 (1)}~ If $\.x$ and $\.y$ are in $V$, then $\.x+\.y$ is defined and in $V$.\\
 {\bf (2)}~ $\.x+(\.y+\.z)=(\.x+\.y)+\.z$\\
 {\bf (3)}~ There is a zero vector $\.0$ with $\.x+\.0=\.x$.\\
 {\bf (4)}~ $\.x+\.y=\.y+\.x$\\
 {\bf (5)}~ If $c$ = constant and $\.x$ is in $V$, then $c \cdot \.x$ is defined and in $V$.\\
 {\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}

\medskip

\begin{minipage}{\textwidth}\small\sf
{\bf Definition}. If vectors $\.b_1,\.b_2,\.b_3$ are a basis for subspace $X$ of an {\em abstract} vector space $V$, and $\.x=c_1\.{b}_1 + c_2\.{b}_2+ c_3\.{b}_3$ is a given linear combination of these vectors, then the uniquely determined constants $c_1,c_2,c_3$ are called the {\em coordinates of $\.{x}$ relative to the
basis} $\.{b}_1, \.{b}_2, \.{b}_3$ and the {\em coordinate map} is the isomorphism
$$\.x=c_1\.{b}_1 + c_2\.{b}_2+ c_3\.{b}_3 \to \colvector{c_1\\ c_2 \\ c_3}.$$
\end{minipage}

\problem{100}{%%%Not on the final exam S2012:
Let $V$ be the vector space of all functions on $(-\infty,\infty)$.
Define $W=\SPAN\{1,x,x^2\}$. Assume known that $1,x,x^2$ are independent functions.
Define subspace $S=\SPAN\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}$ where
$$\vec{v}_1:~ y=1+x,\quad \vec{v}_2:~ y=2+x^2,\quad \vec{v}_3:~ y=2+x+x^2.$$
{\bf (a)} [20\%] Explain why $S$ is contained in $W$, that is, provide details
for why vectors $\vec{v}_1,\vec{v}_2,\vec{v}_3$ are in $W$.

\anotherpage

{\bf (b)} [40\%] Prove that $W=S$. Therefore $\dim(S)=\dim(W)=3$, which proves that $\.v_1,\.v_2,\.v_3$ are independent.

\myspace{10pc}

{\bf (c)} [40\%] Define vector $\.v$ in $S$ by equation $y=3+4x+x^2$.  {\bf Compute} $c_1$, $c_2$, $c_3$ satisfying the equation $\.{v}=c_1\.v1+c_2\.v_2+c_3\.v_3$, using coordinate map methods.

{\bf Expected in (c)}: Calculations of $c_1,c_2,c_3$ are to be done using column vectors from $\Rr^3$, not functions
from $V$. {\bf Zero credit} for not using column vectors.
}{}{~}
%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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~\\
{\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}
\problem{100}{~\newline
{\bf (a)} [40\%] Let $V$ be the vector space $\Rr^4$. Invent an example of a non-void subset $S$ of $V$ that satisfies (1) and (2) but fails the third item (3).

\anotherpage

{\bf (b)} [60\%] Let $V$ be an {\em abstract} vector space.
           Let $\.v_1,\.v_2,\.v_3$ be three vectors in $V$. Define $S$ to be the set of all linear combinations of $\.v_1,\.v_2,\.v_3$. Prove:
           \begin{center} {\bf SPAN THEOREM}. {\sf Subset $S$ is a subspace of $V$}.
           \end{center}

\medskip

{\bf Expected}: {\small\sf A proof uses only the symbols $\.v_1,\.v_2,\.v_3$ and the 8 rules. Column vectors cannot be used in this proof. Please don't try to burst packages $\.x$. }
}{}{~}
%(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.
%
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\problem{100}{
Let $A$ be a $4\times 3$ matrix. Assume the columns of $A^TA$ are independent. Prove or disprove that $A$ has independent columns.
\\[2ex]
{\small\sf {\bf Expected}: To prove the claim, assemble details and theorem citations to support the claim. To disprove the claim, invent a specific $4\times 3$ matrix that violates the claim.}
}{}{~}
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\problem{100}{
Let $U$ be a $2\times 2$ matrix with $U^TU=I$. Let $\.u_1,\.u_2$ denote the columns of $U$. Prove that the columns of $U$ are orthonormal.
}{}{~}
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\problem{100}{
Find the orthogonal projection vector $\.{v}$ (the shadow projection vector) of $\.{y}$ onto the direction of $\.{u}$, given
$$\.{y}=\colvector{1\\ 1\\ 0 \\ 1}, \quad \.{u}=\colvector{1\\ 1\\ 1\\ 0}.$$
}{}{
Use the formula $\vec{v}=d\vec{u}$ where $d=\dd\frac{\vec{y}\cdot\vec{u}}{\vec{u}\cdot\vec{u}}$.
 }
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\problem{100}{
Let $W$ be the column space of $A=\left(\begin{array}{cc}1&1\\1&1\\1&0\end{array}\right)$ and let $\.b=\colvector{1\\ -1\\ 1}$

\medskip

{\bf (a)} [40\%] Define the {\bf normal equations} for the problem $A\.x=\.b$.

\myspace{6pc}

{\bf (b)} [60\%]
Let $\.N$ be the near point to $\.b$ in the subspace $W$. Find $\.N$.
}{}{The columns of $A$ are independent. The normal equation is $A^TA\.{x}=A^T\.{b}$, which in explicit form is $\twobytwo{3&2}{2&2}\.{x} = \colvector{1\\ 0}$.  The answer is $\vec{x}=\colvector{1\\ -1}$. Then
$\.N=A\.{x}=\colvector{0\\ 0\\ 1}$.}

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\problem{100}{
The {\bf Fundamental Theorem of Linear Algebra} contains this statement: {\large\sf The row space of a matrix is orthogonal to the null space of the matrix}. This means that $\.R \cdot \.N = 0$ for each vector $\.R$ in the row space and each vector $\.N$ in the null space.\\[1pc]
Let $A$ be an $m\times n$ matrix. Define subspaces
$S_1$ = column space of $A$,
$S_2$ = null space of $A^T$.
Give a direct proof (using no textbook theorems) that the only vector $\vec{v}$ in both $S_1$ and $S_2$ is the zero vector.
}{}{~}
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