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\examtitle{MATH 2270-003}{Final Exam}{Fall 2010}

\studentinfo
\instructions{You may not use a calculator. Your solutions must include enough justification that another person could understand
 and be convinced by your argument.\\
There are extra blank pages at the end of the booklet. If you need
more room to work a problem please note the page number where your
work continues.\\
There are two Parts. Each Part is worth 50\%.}
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\part{}

\problem{5}{
$L$ is a linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$
such that
$L\begin{pmatrix}1\\1\end{pmatrix}=\begin{pmatrix}1\\0\end{pmatrix}$. What
is $L\begin{pmatrix}2\\2\end{pmatrix}$?
}{}{}

\anotherpage

\problem{5}{
Consider the plane $V$ in $\mathbb{R}^4$ spanned by the vectors
$\bar{v}_1=\begin{pmatrix}1\\1\\1\\0\end{pmatrix}$ and
$\bar{v}_2=\begin{pmatrix}1\\1\\0\\1\end{pmatrix}$. Give an
orthonormal basis for $V$.
}{}{}
\anotherpage

\problem{5}{
$\mathcal{A}=\{\begin{pmatrix}3\\2\end{pmatrix},\begin{pmatrix}4\\4\end{pmatrix}\}$
is a basis for $\mathbb{R}^2$. Let
$\bar{v}=\begin{pmatrix}10\\4\end{pmatrix}$. Write $\bar{v}$ in $\mathcal{A}$--coordinates.
}{}{}

\anotherpage
\problem{5}{
Are the following vectors linearly independent? \[\begin{pmatrix} 1 \\ 2 \\ 3\end{pmatrix}, \begin{pmatrix} 4\\ 0\\ 4\end{pmatrix}, \begin{pmatrix} 3 \\ 2
  \\1\end{pmatrix}\]
}{}{}

\anotherpage


\problem{5}{
Use least squares approximation to find the best fit line for the
points $(1,2)$, $(2, 2)$, $(3,0)$.
}{}{}
\anotherpage

\problem{5}{
Find all solutions to the following system of equations:
\begin{align*}
  2w+3x+4y+5z = 1\\
 4w+3x+8y+5z = 2\\
 6w+3x+8y+5z = 1\\
\end{align*}
}{}{}

\anotherpage

\problem{5}{
Find an orthogonal matrix $Q$ and diagonal matrix $D$ such that
$A=QDQ^T$ for $A=\begin{pmatrix}3 & -1\\ -1 &3\end{pmatrix}$.
}{}{}

\anotherpage

\problem{5}{
Show that if $B$ is an invertible matrix and $A$ is similar to $B$
then $A$ is invertible.
}{}{}

\anotherpage

\problem{5}{
$M$ is an $m\times n$ matrix. Prove that the null space of $M$ is a
vector subspace of $\mathbb{R}^n$.
}{}{}

\anotherpage
\problem{5}{
$A$ is an $m\times n$ matrix of rank $r$. Show that the rank of $A^TA$
is also $r$. Conclude that $A^TA$ is invertible if and only if the
columns of $A$ are linearly independent.
}{}{}
\anotherpage




\anotherpage
\part{}
State the Fundamental Theorem of Linear Algebra, Parts 1 and 2, and
the Singular Value Decomposition (SVD) Theorem. Explain, in plain English,
what they mean and how they complement each other. Give an explicit
(actual numbers in the entries)
example of a matrix $M$ such that each of the four fundamental
subspaces is nontrivial, and such that the two bases in the SVD are
not the same. Compute the SVD of $M$ and identify orthonormal bases for
the four fundamental subspaces.





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