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\begin{document}


\begin{center}
{\LARGE {Mathematics 3A, Section A, Winter 1997}}
\end{center}
\begin{center}
\large Midterm Exam I
\end{center}

\begin{enumerate}

\item Solve the system of linear equations
\begin{equation*}
\begin{pmatrix}
  -2 & 1  & -5 & -1 \\
  1  & 0  & 2  & 0  \\
  1  & 1  & 1  & 1  \\
  -1 & 2  & -4 & -1
\end{pmatrix}
\begin{pmatrix}
  x_1 \\ x_2 \\ x_3 \\ x_4
\end{pmatrix} =
\begin{pmatrix}
  -2 \\ -1 \\ 9 \\ 0
\end{pmatrix}.
\end{equation*}
Use the method taught in class.  Check your answer in a meaningful way
(required!).

\hfill (30)

\item Let $V$ be a vector space, and $\{ v_1, \ldots, v_4 \} \subset
V$ a set of linearly independent vectors (so $\dim V \geq 4$).  Let
\begin{alignat*}{2}
  w_1 &= 3 \, v_1 && + v_2 - v_3 - 3 \, v_4 \\
  w_2 &= -2 \,v_1 && - v_2 + v_3 + 2 \, v_4 \\
  w_3 &= -v_1     && - v_2 + v_3 + v_4 \\
  w_4 &=          && - v_2 + v_3
\end{alignat*}
\begin{enumerate}
\item Use coordinazation to show that $w_1, \ldots, w_4$ are linearly
dependent.
\item Find a basis for $\Span \{ w_1, \ldots, w_4 \}$.
\end{enumerate}
\hfill (10+5)

\item Let $V$ be a vector space, and $W \subset V$ a subspace of $V$.
Prove the following.
\begin{enumerate}
\item $0 \in W$.
\item The complement of $W$, i.e.\ the set
\begin{equation*}
  V \backslash W = \{ v \in V \colon v \notin W \},
\end{equation*}
is not a subspace.
\item If $\dim W = \dim V$, then $W = V$.
\end{enumerate}
\hfill (5+5+5)


\item Let $A \in M(m \times n)$ with $m > n$.  Write
\begin{equation*}
  A = \begin{pmatrix} B \\ C \end{pmatrix},
\end{equation*}
where $B \in M(n \times n)$ and $C \in M((m-n) \times n)$.
\begin{enumerate}
\item Prove that if $B$ is invertible, then $\rank A = n$.
\item Give an example that the converse of the statement in (b), i.e.\
the statement that $\rank A = n$ implies $B$ is invertible, is false.
\end{enumerate}
\hfill (10+5)

\item Let $S \colon \mathbb{R}^n \to \mathbb{R}^n$ and $T \colon
\mathbb{R}^n \to \mathbb{R}^n$ be linear transformations.
\begin{enumerate}
\item Show that their composition $T \circ S \colon \mathbb{R}^n \to
\mathbb{R}^n$ defined by $(T \circ S)(x) = T (S (x))$ is also a linear
transformation.

\item Prove that if $T \circ S$ is invertible, then $T$ and $S$ are
both invertible.

\item Let $S \colon \mathbb{R}^2 \to \mathbb{R}^2$ and $T \colon
\mathbb{R}^2 \to \mathbb{R}^2$ be defined by
\begin{equation*}
S(\boldsymbol{x}) =
\begin{pmatrix}
  x_1 + x_2 \\ 2 \, x_1 + x_2 
\end{pmatrix}, \qquad T (\boldsymbol{x}) =
\begin{pmatrix}
  x_1 - x_2 \\ x_1 + x_2
\end{pmatrix}.
\end{equation*}
Give the standard matrix representations for $T \circ S$ and $(T \circ
S)^{-1}$
\end{enumerate}
\hfill (5+10+10)

  
\end{enumerate}

\end{document}
