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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 4}
\date{due March 3, 2004, before 12:00}
\maketitle

\noindent
Normal questions and advanced questions (A) are worth $5$ points; easy
questions (E) are worth $4$ points.  Complete either the easy, or the
advanced version, not both.

\begin{enumerate}
\item Consider the vector space of functions that is spanned by the
basis
\[
  B = \{ \sin x, \cos x, \sin 2 x, \cos 2 x \} \,.
\]
Find the matrix representing the derivative operator with respect to
the basis $B$.

\item Recall the definitions of range and kernel of a linear map $A$
on the vector space $\R^n$:
\begin{gather*}
  \Range A = \{ A \x \colon \x \in \R^n \} \\
  \Ker A = \{ \x \in \R^n \colon A\x = 0 \}
\end{gather*}
\begin{itemize}
\item[(E)] Let
\[
  A = \begin{pmatrix}
        -1 & 2 & -2 \\
        1 & 2 & 6 \\
        0 & 1 & 1
      \end{pmatrix}
\]
Find a basis for $\Range A$ and for $\Ker A$.

\item[(A)] Prove that $\dim \Ker A + \dim \Range A = n$.

\end{itemize}

\item Let
\[
  A = \begin{pmatrix}
        2 & 0 & 2 \\
        0 & 2 & 0 \\
        -2 & 0 & 2     
      \end{pmatrix}
\]
represent a linear transformation on $\R^3$ with respect to the
standard basis $\{ \e_1, \e_2, \e_3 \}$.  Find the matrix $A'$ which
represents this transformation with respect to the new basis
\[
  \e_1' =
  \begin{pmatrix}
  0 \\ 1 \\ 1
  \end{pmatrix} \,, \quad
  \e_2' =
  \begin{pmatrix}
  1 \\ 0 \\ 1
  \end{pmatrix} \,, \quad
  \e_3' =
  \begin{pmatrix}
  1 \\ 1 \\ 0
  \end{pmatrix} \,.
\]

\item Compute the determinant
\[
  \begin{vmatrix}
    1 & 2 & 0 & 0 \\
    1 & 1 & 0 & 0 \\
    0 & 0 & 7 & 3 \\
    0 & 0 & 1 & 1
  \end{vmatrix} \,.
\]

\item
\begin{itemize}
\item[(E)] Use the determinant to test if the matrix
\[
  A = \begin{pmatrix}
  2 & -1 & 0 \\
  -1 & 2 & -1 \\
  1 & -1 & 1
\end{pmatrix} 
\]
represents an invertible linear transformation.

\item[(A)] Use the definition of the determinant to show that a matrix
$A$ is invertible if and only if and only if $\det A \neq 0$.

\end{itemize}

\item Show that a matrix $A$ is invertible if and only if all the
eigenvalues of $A$ are nonzero.

(Recall that $\lambda$ is an eigenvalue of $A$ and $\v$ is the
corresponding eigenvector if $A\v=\lambda \v$.)



\end{enumerate}

\end{document}
