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\allowdisplaybreaks % fixes align environment weird spacing on page
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%For references using Lemma, Theorem, etc, use \cref
\usepackage[nameinlink,capitalize,sort]{cleveref}

% theorems
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
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\AtBeginDocument{\xpatchcmd{\@thm}{\thm@headpunct{.}}{\thm@headpunct{}}{}{}}



% lin alg
\newcommand{\bu}{{\mathbf{u}}}
\newcommand{\bv}{{\mathbf{v}}}
\newcommand{\bw}{{\mathbf{w}}}
\newcommand{\bx}{\mathbf{x}}
\newcommand{\zerovec}{{\mathbf{0}}}

% other useful stuff
\newcommand{\Id}{{\mathds{1}}}
\newcommand{\R}{{\mathbb{R}}}
\newcommand{\C}{{\mathbb{C}}}
\newcommand{\Z}{{\mathbb{Z}}}
\newcommand{\N}{{\mathbb{N}}}
\newcommand{\Q}{{\mathbb{Q}}}
\newcommand{\F}{{\mathbb{F}}}
\newcommand{\cL}{{\mathcal{L}}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\inv}{{-1}}

%\DeclareMathOperator{\dim}{dim}
\DeclareMathOperator{\range}{range}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\nullspace}{null}
\DeclareMathOperator{\nullity}{nullity}


\begin{document}
\begin{center}
\Large{Exercises for Module 7: Linear Algebra I}
\end{center}

1. Suppose that $\alpha\in\F, \bv\in V$, and $\alpha \bv=\zerovec$. Prove that $a=0$ or $\bv=0$.

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2.  Prove the following:  Let $V$ be a vector space and let $U_1, U_2 \subseteq V$ be subspaces. Then $U_1 \cap U_2$ is also a subspace of $V$. 

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3. Let $U_1$ and $U_2$ be subspaces of a vector space $V$. Prove that $U_1 \cup U_2$ is a subspace of $V$ if and only if $U_1 \subseteq U_2$ or $U_2 \subseteq U_1$.
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4.  Suppose $\bv_{1},...,\bv_{m}$ is linearly independent in $V$ and $\bw\in V$. Prove that if $\bv_{1}+\bw,...,\bv_{m}+\bw$ is linearly dependent, then $\bw\in\text{span}(\bv_{1},...,\bv_{m})$. 

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5. Suppose that $\bv_{1},...,\bv_{m}$ is linearly independent
in $V$ and $\bw\in V$. Show that $\bv_{1},...,\bv_{m},\bw$ is linearly independent if and only if 
$$\bw\notin\text{span}\{\bv_{1},...,\bv_{m}\}$$

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6. Let $T \in \cL(\mathbb{P}(\R),\mathbb{P}(\R))$ be the map $T (p(x))= x^2 p(x)$ (multiplication by $x^2$). 
\begin{enumerate}
    \item[(i)] Show that $T$ is linear.
    \item[(ii)] Find the null space and range of $T$.
\end{enumerate}

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7. Let U and V be finite-dimensional vector spaces and $S \in \mathcal{L}(V,W)$ and $T \in \mathcal{L}(U,V)$. Show that 
\begin{equation*}
    \dim \nullspace ST \leq \dim \nullspace S + \dim \nullspace T
\end{equation*}
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\end{document}


