HW Problem $7$

Let

$A\left[\begin{array}{c}6\\ 0\\ -3\end{array}\right]=\left[\begin{array}{c}4\\ -2\\ -5\end{array}\right],A\left[\begin{array}{c}3\\ -3\\ 3\end{array}\right]=\left[\begin{array}{c}1\\ -5\\ 1\end{array}\right],$ and $,A\left[\begin{array}{c}3\\ 3\\ 0\end{array}\right]=\left[\begin{array}{c}-1\\ -1\\ -4\end{array}\right]$

$($i$)(1$ pt$.)$ What are the dimensions of $A?$

$($ii$)(8$ pts$.)$ Determine the three columns $a^{(1)},a^{(2)}$ and $a^{(3)}$ of $A($and thus A itself$).$ Hint: Sum

both sides of the last two equalities.

$($iii$)(4$ pts$.)$ In the usual three$-$dimensional Cartesian space $($with orthogonal axes$),$ consider the

plane $S$ which contains the origin and has normal vector ${\left[\begin{array}{ccc}1& 1& 1\end{array}\right]}^T.$ Determine the reflection of the

point ${\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T$ about $S.($Hint: Express ${\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T$ as a sum of two vectors, one parallel to ${\left[\begin{array}{ccc}1& 1& 1\end{array}\right]}^T$

and one orthogonal to it$.)$

$($iv$)(4$ pts$.)$ By symmetry, determine the reflections of ${\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T$ and ${\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T$ about $S.$ Explain

why the matrix A represents a reflection of an arbitrary point $($in three$-$dimensional space$)$ about

$S.$

$($v$)(3$ pts$.)$ Study the function TOEPLITZ in MATLAB documentation. Give a single row vector

c such that

$A=$ toeplitz$(c)$

produces the matrix $A$ of parts $($i$)-($iv$)$ above.