\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts,graphicx}

\def\d{{\mathrm d}}
\def\e{{\mathrm e}}
\def\R{{\mathbb R}}
\def\ba{\boldsymbol{a}}
\def\bb{\boldsymbol{b}}
\def\bc{\boldsymbol{c}}
\def\bd{\boldsymbol{d}}
\def\bn{\boldsymbol{n}}
\def\bu{\boldsymbol{u}}
\def\bv{\boldsymbol{v}}
\def\bw{\boldsymbol{w}}
\def\bp{\boldsymbol{p}}
\def\bx{\boldsymbol{x}}

\begin{document}

\title{Calculus and Elements of Linear Algebra I}
\author{Homework 11}
\date{Due on Moodle, Monday, November 30, 2020}
\maketitle


\begin{enumerate}

\item 
Solve the following system of linear equations using the method
taught in class.
\begin{align*}
  x_1 + 3 \, x_2 - 5 \, x_3 &= 4 \\
  x_1 + 4 \, x_2 - 8 \, x_3 &= 7 \\
  -3 \, x_1 - 7 \, x_2 + 9 \, x_3 &= -6
\end{align*}

\item Find conditions on $\alpha$ such that following system of
linear equations has (a) exactly one solution, (b) no solutions, or
(c) an infinite number of solutions; give all solutions where they
exist.
\begin{align*}
  x_1 + \alpha \, x_2  &= 1 \\
  x_1 - x_2 + 3 \, x_3 &= -1 \\
  2 \, x_1 -2 \, x_2 + \alpha \, x_3 &= -2
\end{align*}



\item Let $\bv=(1,2,3)^T$ 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
\[
  \bb_1 =
  \begin{pmatrix}
  0 \\ 1 \\ 1
  \end{pmatrix} \,, \quad
  \bb_2 =
  \begin{pmatrix}
  1 \\ 0 \\ 1
  \end{pmatrix} \,, \quad
  \bb_3 =
  \begin{pmatrix}
  1 \\ 1 \\ 0
  \end{pmatrix} \,.
\]



\item 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.
\[
  \bv_1 = 
  \begin{pmatrix}
  1\\0\\0\\1
  \end{pmatrix}\,, \quad
  \bv_2 = 
  \begin{pmatrix}
  1\\-1\\-1\\2
  \end{pmatrix}\,, \quad
  \bv_3 = 
  \begin{pmatrix}
  0\\1\\-1\\1
  \end{pmatrix}\,, \quad
  \bv_4 = 
  \begin{pmatrix}
  -1\\3\\1\\0
  \end{pmatrix}\,.
\]

\item A matrix is called \emph{singular} if the homogeneous linear
system $A \bv = \bb$ has a ``non-trivial'' solution $\bv \neq 0$.

Prove that $AB=0$ implies that at least one of the matrices
is singular.

\end{enumerate}

\end{document}
