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\lhead{{\sf Summer Bridge Course (Algebra)}: Tuesday July 7}
\chead{}\rhead{\thepage}
\lfoot{Due Wednesday, July 8 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}


\begin{document}

%$\;$ \vspace{-4mm}


\noindent \textbf{Topics}: Orthogonality, Gram-Schmidt, and QR factorization \\

\noindent \textbf{Do}: Answer the following questions.

%%--------------------------------------------------------------------- 

  % Strang: Section 4.4, #10 (modified)
  \begin{exercise}
    In this problem you will prove that orthonormal vectors are linearly
    independent two different ways.
    \begin{enumerate}[label=(\roman*)]
    \item Vector proof: First, suppose that
      $c_1\q_1+c_2\q_2+\cdots+c_k\q_k=\0$. Show that each
      $c_i=0$. [\emph{Hint}: Start by multiplying both sides of the
      equation by $\q_i^T$.]
    \item Matrix proof: Let $\bold{Q}$ be the matrix whose columns are the
      $\q_i$'s. Show that if $\bold{Q}\x=\0$, then $\x=\0$. [\emph{Hint}:
      Since $\bold{Q}$ need not be square, you cannot assume $\bold{Q}^{-1}$
      exists, but $\bold{Q}^T$ of course will.] \\
    \end{enumerate}
  \end{exercise}
    
%%--------------------------------------------------------------------- 
  
  % Strang: Section 4.4, #32, 34 (modified)
  \begin{exercise}
    Recall that if $||\u||=1$, then the rank-$1$ matrix $\u\u^T$ is
    the projection matrix onto $\u$. In this case, $\bold{Q}=\I-2\u\u^T$ is a
    \emph{reflection matrix}.
    \begin{enumerate}[label=(\roman*)]
    \item Reflecting twice across the same axis is the identity. Verify
      that indeed, $\bold{Q}^2=\I$. 
    \item Compute $\bold{Q}\u$, and simplify this expression as much as
      possible.
    \item Suppose $\v$ is orthogonal to $\u$. Compute $\bold{Q}\v$, and
      simplify as much as possible.
    \item Describe in plain English which subspace $\bold{Q}$ is reflecting
      across. Your answer should involve $\u$. Include a sketch. 
    \item Compute the reflection matrix $\bold{Q}_1=\I-2\u_1\u_1^T$ where
      $\u_1=(0,1)$. Compute $\bold{Q}_1\x_1$, where $\x_1=(a,b)$, and sketch
      the vectors $\u_1$, $\x_1$, and $\bold{Q}_1\x_1$ in the plane.
    \item Compute the reflection matrix $\bold{Q}_2=\I-2\u_2\u_2^T$ where
      $\u_2=(0,\sqrt{2}/2,\sqrt{2}/2)$. Compute $\bold{Q}_2\x_2$, where
      $\x_2=(1,1,1)$, and sketch the vectors $\u_2$, $\x_2$, and
      $\bold{Q}_2\x_2$ in $\R^3$. \\
  \end{enumerate}
  \end{exercise}
  
%%--------------------------------------------------------------------- 
  
  % Strang: Section 4.4, #18 (modified)
  \begin{exercise}
    Let $\a$, $\b$, and $\c$ be the (independent) column vectors of
    the matrix
  \[
  \bold{M}=\begin{bmatrix}1&0&0\\-1&1&0\\0&-1&1\\0&0&-1\end{bmatrix},
  \]
  whose column space is the orthogonal complement to the vector $(1,1,1,1)$. Use the Gram-Schmidt process to construct an orthonormal basis $\q_1$, $\q_2$, and $\q_3$. Then write $\bold{M}=\bold{Q}\bold{R}$, where $\bold{Q}$ is orthogonal and $\bold{R}$ is upper-triangular. \\
  \end{exercise}

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\end{document}