\documentclass[10pt,reqno,letterpaper]{amsart}

\usepackage{tabularx}
\usepackage{enumitem}
\usepackage{url}

\newcommand{\RR}[0]{\mathbb{R}}
\newcommand{\NN}[0]{\mathbb{N}}
\newcommand{\FF}[0]{\mathbb{F}}
\newcommand{\CC}[0]{\mathbb{C}}
\newcommand{\ZZ}[0]{\mathbb{Z}}
\newcommand{\spn}[0]{\operatorname{span}}

\pdfpagewidth 8.3in
\pdfpageheight 11.7in
\setlength\textwidth{6.15in}
\setlength\oddsidemargin{0.1in}
\setlength\evensidemargin{0.1in}

%\pagenumbering{gobble}
\title{MATH 202A \\ Applied Algebra I \\ Fall 2019}

\begin{document}

\maketitle

\section*{Homework week 1}
\vspace{0.5\baselineskip}

Due by the beginning of class on Friday 4th October (hand in via Gradescope).
\vspace{0.5\baselineskip}

\begin{enumerate}[label={\bf\arabic*.}, itemsep=0.8\baselineskip, leftmargin=2em, labelsep=2em,parsep=0.4\baselineskip]

  \item Prove, using only the results from Section 1.2, that (i) any finite-dimensional vector space has a basis, and (ii) any two bases of a finite dimensional vector space $V$ have the same size.

    [Recall that we say a vector spaces is finite-dimensional to mean that it has a finite spanning set.]

  \item Let $V$ be a finite dimensional vector space over the finite field $\FF_p$ ($p$ prime).  Show that $|V|$, the number of distinct vectors in $V$, is finite and a power of $p$.

  \item Let $V$, $W$ be vector spaces, $U \subseteq V$ a subspace, and $\phi \colon V \to W$ a linear map.  Suppose also that $\ker \phi \supseteq U$.

    Prove that there exists a unique linear map $\psi \colon V/U \to W$ such that $\phi(v) = \psi(v + U)$ for all $v \in V$.

    Prove that $\psi$ is injective if and only if $U = \ker \phi$.

  \item Let $V$ be a finite-dimensional vector space and $U \subseteq V$ a subspace.  Suppose that $u_1,\dots,u_k$ are a basis for $U$, and $v_1,\dots,v_m$ are vectors in $V$ such that $v_1+U,\dots,v_m+U$ is a basis for $V/U$.

    Prove that $u_1,\dots,u_k,v_1,\dots,v_m$ is a basis for $V$ (and hence $\dim V = \dim U + \dim (V/U)$).

  \item Suppose $V$ is a vector space over $\RR$, $w_1, w_2, w_3 \in V$ span $V$ and $u_1,u_2,u_3$ are linearly independent where
    \begin{align*}
      u_1 &= 4 w_1 -3 w_3 \\
      u_2 &= -w_1 + w_3 \\
      u_3 &= w_1 + w_2 + w_3 \, .
    \end{align*}
    By Theorem 1.2.1, we can add $3-3 = 0$ vectors from $w_1,w_2,w_3$ to $u_1,u_2,u_3$ to make a spanning set; i.e.~$u_1, u_2, u_3$ already spans $V$.   In particular there exist coefficients $a_1, a_2, a_3 \in \RR$ such that $a_1 u_1 + a_2 u_2 + a_3 u_3 = w_3$.

    Our proofs are constructive.  So, we can (and will) run the proof of Theorem 1.2.1 for $k=m=3$ and these $u_i, w_i$, line by line, to compute algorithmically what the scalars $a_1,a_2,a_3$ are.  [For this question, don't compute them ``by hand'': follow the proof as directed.]

    \begin{enumerate}[label={\bf(\alph*)}, leftmargin=0em, labelsep=0.5em, itemsep=0.5\baselineskip]
      \item Consider the list $u_1,w_1,w_2,w_3$. It is linearly dependent (by Lemma 1.2.3, as $w_1,w_2,w_3$ spans).  Use the proof to find a dependence relation among these vectors.  Hence (as in Lemma 1.2.2) find an expression for $w_3$ as a linear combination of $u_1,w_1,w_2$.
      \item Using (a), express $u_2, u_3$ as linear combinations of $u_1, w_1, w_2$.
      \item Consider the list $u_2,u_1,w_1,w_2$.  It is linearly dependent (by Lemma 1.2.3, as $u_1,w_1,w_2$ spans).  Use the proof, and (b), to find a dependence relation among these vectors.  Hence (as in Lemma 1.2.2) find an expression for $w_1$ as a linear combination of $u_1,u_2$.
      \item Using (b) and (c), express $u_3$ as a linear combination of $u_1, u_2, w_2$.
      \item Consider the list $u_3,u_2,u_1,w_2$.  It is linearly dependent (by Lemma 1.2.3).  Use the proof, and (d), to find a dependence relation among these vectors.  Hence (as in Lemma 1.2.2) find an expression for $w_2$ as a linear combination of $u_1,u_2,u_3$.
      \item Using your expressions for $w_1$ from (c), $w_2$ from (e), and $w_3$ from (a), write $w_3$ as a linear combination of $u_1,u_2,u_3$.
    \end{enumerate}

    Note the similarity of this process with any other algorithms (Gaussian elimination / reducing to RREF / LU decomposition / etc.) with which you are familiar.  [You need not write anything for this part.]
\end{enumerate}

\end{document}
