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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 3}
\date{due February 22, 2003, before 24:00}
\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
Determine if the following are vector spaces.  If not, explain which
property fails.
\begin{enumerate}
\item The polynomials of degree smaller or equal to $n$, with the
usual addition and multiplication by a scalar.
\item The polynomials of degree $n$, with the usual addition and
multiplication by a scalar.
\item The set of $n\times m$ matrices with the usual addition and
multiplication by a scalar.
\item The set of $n\times m$ matrices with matrix multiplication
taking the role of vector addition, scalar multiplication being as
usual.
\item The set of symmetric $n \times n$ matrices with the usual
addition and multiplication by a scalar.
\item The set of invertible $n \times n$ matrices with the usual
addition and multiplication by a scalar.
\end{enumerate}


\item Let $\v=(1,2,3)^T$ be a vector expressed in coordinates with
respect to the standard basis of $\mathbb R^3$.  Find the coordinates
of this vector with respect to the basis
\[
  \b_1 =
  \begin{pmatrix}
  0 \\ 1 \\ 1
  \end{pmatrix} \,, \quad
  \b_2 =
  \begin{pmatrix}
  1 \\ 0 \\ 1
  \end{pmatrix} \,, \quad
  \b_3 =
  \begin{pmatrix}
  1 \\ 1 \\ 0
  \end{pmatrix} \,.
\]

\item 
\begin{itemize}
\item[(E)] Determine whether the following vectors form a basis of
$\mathbb R^4$.  If not, obtain a basis by adding and/or removing
vectors from the set.
\[
  \v_1 = 
  \begin{pmatrix}
  1\\0\\0\\1
  \end{pmatrix}\,, \quad
  \v_2 = 
  \begin{pmatrix}
  1\\-1\\-1\\2
  \end{pmatrix}\,, \quad
  \v_3 = 
  \begin{pmatrix}
  0\\1\\-1\\1
  \end{pmatrix}\,, \quad
  \v_4 = 
  \begin{pmatrix}
  -1\\3\\1\\0
  \end{pmatrix}\,.
\]

\item[(A)] Let $V$ be a vector space, $\b_1,\dots,\b_n$ a basis of
$V$, and $\v_1,\dots,\v_m$ where $m\leq n$ a set of linearly
independent vectors in $V$.  Show that you can construct another basis
for $V$ consisting of $\v_1,\dots,\v_m$ and $n-m$ vectors from among
$\b_1, \dots,\b_n$.

Hint: Successively replace one of the $\b_i$ by a vector $\v_j$.
\end{itemize}

\item Use the definition of the matrix inverse to show that
$(AB)^{-1}=B^{-1}A^{-1}$.

\item Use the method taught in class to compute the inverse of
\[
\begin{pmatrix}
  2 & -1 & 0 \\
  -1 & 2 & -1 \\
  0 & -1 & 2
\end{pmatrix} \,.
\]

\item 
\begin{itemize}
\item[(E)] Is the following matrix invertible?  If yes, compute its
inverse.
\[
  \begin{pmatrix}
  0 & -1 & 0 & 0 \\
  1 & 0 & 0 & 0 \\
  0 & 0 & 0 & -1 \\
  0 & 0 & 1 & 0
  \end{pmatrix} \,.
\]

\item[(A)] Prove that the following are equivalent.
\begin{enumerate}
\item $A\in M(n\times n)$ is invertible.
\item $\operatorname{Ker}A=\{\v \in \mathbb R^n \colon A\v=0\}$
contains only the zero vector.
\item The columns of $A$ are linearly independent.
\end{enumerate}

\end{itemize}


\end{enumerate}

\end{document}
