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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Review for Midterm I}
\date{March 7, 2003, 11:00--12:15}
\maketitle

\begin{enumerate}

\item Equations for lines and planes; distance of a point to a line or
plane; distance between two lines.  See, in particular, the examples
on pp.\ 234--237.

\item Complex Numbers: Know how to do arithmetic with complex numbers;
polar representation of complex numbers; complex logarithm.

\item Solve a system of linear equations: See handout.  Practice
problem:
\[
  A = 
  \begin{pmatrix}
  1 &2 &5 &1 &0 \\ - 1& - 1& - 4& 1& 1\\
  0 &- 1 &- 1 &- 1& 0\\ 1& 2& 5& 0&-  1
  \end{pmatrix} \,, \qquad
  \b =
  \begin{pmatrix}
  0\\1\\0\\-1
  \end{pmatrix} \,.
\]
\begin{enumerate}
\item Solve $A\x=\b$.  \item Characterize all vectors $\b$ for which
the equation has a solution.
\item Find a basis for the kernel of $A$.
\item Find a basis for the range of $A$.
\end{enumerate}

\item Concept of vector space, linear independence, basis. 

\item Matrix inversion: see handout and examples from homework.

\item Linear transformations: Definition, representation by a matrix,
change of basis.  Practice problem:

Let
\[
  \v_1 = 
  \begin{pmatrix}
  2 \\-2 \\-1
  \end{pmatrix} \,, \quad
  \v_2 = 
  \begin{pmatrix}
  2 \\1 \\-2
  \end{pmatrix} \,, \quad
  \v_3 = 
  \begin{pmatrix}
  3 \\-1 \\-2
  \end{pmatrix} \,,
\]
and 
\[
  \w_1 = 
  \begin{pmatrix}
  -1 \\ -1 \\ -1
  \end{pmatrix} \,, \quad
  \w_2 = 
  \begin{pmatrix}
  0 \\ 3 \\ -2
  \end{pmatrix} \,, \quad
  \w_3= 
  \begin{pmatrix}
  1 \\ -1 \\ 2
  \end{pmatrix} \,,
\]
be two bases in $\R^3$.
\begin{enumerate}
\item Write the vector $\u=(2,3,5)^T$ in the basis $\{\v_1, \v_2, \v_3\}$.
\item Find the change of basis matrix from the basis $\{\v_1, \v_2,
\v_3\}$ to the basis $\{\w_1, \w_2, \w_3\}$.
\item Let $\a = 4\v_1-2\v_2+\v_3$; find its coordinates in the basis
$\{\w_1, \w_2, \w_3\}$.
\item Let the linear transformation $F$ be the reflection about the
plane spanned by the standard unit vectors $\e_1$ and $\e_2$; find the
matrix representing $F$ in the standard basis and in the basis
$\{\v_1, \v_2, \v_3\}$.
\item Compute $F(\a)$ in the basis $\{\v_1, \v_2, \v_3\}$.
\item Let $\b = \w_1+2 \w_2 + 3 \w_3$; compute $F(\b)$ in the basis
$\{\v_1, \v_2, \v_3\}$.
\end{enumerate}

\item Determinants.  See, for example, p.\ 312 question 8.2.  Part (b)
was done in class, the solution given in the book is wrong!

\item Eigenvalues and eigenvectors.  Practice problem: Diagonalize the
matrix
\[
  B =
  \begin{pmatrix}
  0 & -i & i \\ i & 0 & -i \\ -i & i & 0
  \end{pmatrix} \,.
\]

\item Test your understanding:  Are the following statements true or
false?  If false, give a short argument.
\begin{enumerate}
\item If $A$ is an $n\times   k$ matrix with $n > k$, then its columns
are linearly independent.
\item If $A$ is an $n\times   k$ matrix with $n < k$, then its columns
 are linearly dependent.
\item $A\x = \b$ has infinitely many solutions if the nullspace of $A$
is nontrivial.
\item Suppose that $A$ is invertible. Then $A^T$ is also invertible
and its inverse is the transpose of $A^{-1}$.
\item There exists such a $3\times   3$ matrix $A$ that $\Range A =
\Ker A$
\item The set of orthogonal $3 \times 3$ matrices forms a vectorspace
with the usual matrix addition and scalar multiplication.  (Recall
that a matrix is orthogonal if $A^T=A^{-1}$.) 

\item The projection onto the plane $x + 3y - 2z = 1$ in $\R^3$ is a
linear transformation.

\item Two eigenvectors of a matrix $A$ are always linearly
independent.

\item Every square matrix is diagonalizable.

\item Every regular square matrix is diagonalizable

\item Every hermitian matrix is diagonalizable.
\end{enumerate}

\end{enumerate}



\end{document}
