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%Fill in the appropriate information below
\lhead{Your name: }
\rhead{Math 457 Fall 2021}
\chead{\textbf{LA Homework 1}}
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\begin{document}
\begin{mdframed}[backgroundcolor=blue!20]
\LaTeX submissions are mandatory. The template for this problem can be found on the Piazza resource page for this course.
\end{mdframed}
\begin{problem}{}
Let B be a 4 x 4 matrix to which we apply the following operations:
1. double column 1,
2. halve row 3,
3. add row 3 to row 1,
4. interchange columns 1 and 4,
5. subtract row 2 from each of the other rows,
6. replace column 4 by column 3,
7. delete column 1 (so that the column dimension is reduced by 1).
(a) Write the result as a product of eight matrices.
(b) Write it again as a product ABC (same B) of three matrices.
\end{problem}
\begin{solution}
\end{solution}
\begin{problem}{}
Write the matrix $\bigg(\Big((AB)^t\Big)^{-1}\bigg)^t$ in terms of $A^{-1}$ and $B^{-1}$.
\end{problem}
\begin{solution}
\end{solution}
\begin{problem}{}
Let $X$ be a matrix
\[
X =
\bmatr{
A& B \\
C& D
}.
\]
where $A$ and $D$ are $n\times n$ and $m\times m$ matrices, respectively, and suppose that $A$ is invertible.
The \emph{Schur complement} matrix $S$ is defined through the formula
\[
\bmatr{
I& 0 \\
-CA^{-1} & I
}
\bmatr{
A& B \\
C & D
}
=
\bmatr{
A& B \\
0 & S
}
\]
Express $S$ in terms of matrices $A$, $B$, $C$ and $D$. What are the dimensions of $S$?
\end{problem}
\begin{solution}
\end{solution}
\begin{problem}{}
By using the reduction to the rref form, find the bases for the column space and nullspace of $A$ and the solution to $Ax = b$:
\[
A =\bmatr{
2 & 4 & 6 & 4 \\
2 & 5 & 7 & 6 \\
2 & 3 & 5 & 2
}
\quad
b = \bmatr{
4 \\ 3 \\ 5
}
\]
\end{problem}{}
\begin{solution}
\end{solution}
\newpage
\begin{problem}{}
Let $f_1, \ldots f_8$ be a set of functions defined on the interval $[1, 8]$ with the property that for any numbers $d_1, \ldots , d_8$, there exists a set of coefficients $c_l, \ldots , c_8$ such that
\bal{
\sum_{j = 1}^{8} c_j f_j(i) = d_i, \quad i = 1, \ldots, 8.
}
(a) Show by appealing to the theorems of lecture 1 in Trefethen, Bau that $d_1, \ldots , d_8$ determine $c_1, \ldots, c_8$ uniquely.
(b) Let $A$ be the $8 \times 8$ matrix representing the linear mapping from data $d_1, \ldots , d_8$ to coefficients $c_1, \ldots , c_8$. What is the $i, j$ entry of $A^{-1}$?
\end{problem}
\begin{solution}
\end{solution}
\end{document}