7-18.tex

\documentclass{article}

\usepackage{fullpage}

\title{Midterm Review}
\date{2018-07-18}
\begin{document}
\maketitle

Throughout, let $V,W$ be vector spaces over a field $F$ and $T:V\to W$ a linear
map.
\begin{enumerate}
    \item
        Prove that the intersection of 2 subspaces is a subspace.
    \item
        Let $V_1,V_2\subseteq V$ be subspaces. Find necessary and sufficient
        conditions for $V_1\cup V_2$ to be a subspace.
    \item
        Let $V_1,V_2\subseteq V$ be subspaces. Find necessary and sufficient
        conditions for $V_1\setminus V_2$ to be a subspace.
    \item
        Let $V_1,V_2\subseteq V$ be subspaces. Find necessary and sufficient
        conditions for $V_1+ V_2=\{v_1+v_2:v_1\in V_1, v_2\in V_2\}$ to be a
        subspace.
    \item
        Prove that $V_1+V_2:=\{v_1+v_2:v_1\in V_1, v_2\in V_2\}$ is the
        smallest subspace of $V$ containing both $V_1$ and $V_2$.
    \item
        Prove that $V\times W$ is a vector space with the addition law and
        scalar multiplication law derived from the addition law and scalar
        multiplication law from $V$ and $W$ so
        \[
            (v_1, w_1)+(v_2,w_2)=(v_1+v_2, w_1+w_2), \quad
            c(v_1,w_1)=(cv_1,cw_1).
        \]
    \item
        Suppose $V_1, V_2$ are subspaces of a finite dimensional space $V$, prove
        \[
            \dim V_1+\dim V_2 -\dim V \leq \dim (V_1\cap V_2).
        \]
    \item
        Prove that $S=\{p\in P_5(F):p''+2p'=0\}$ is a subspace of $P_5(F)$.
        What is $\dim S$?
    \item
        Prove that $S = \{A\in M_n(F):\mathrm{tr}(A)=0\}$ is a subspace
        of $M_n(F)$. What is $\dim S$?
    \item
        Is the set of invertible $n\times n$ matrices a subspace?
    \item
        Is the set of symmetric $n\times n$ matrices a subspace?
    \item
        Is the set of $3\times 3$ rank 2 matrices a subspace?
    \item
        Suppose $T:V\to W$ and $S:W\to V$ are linear maps so that $S\circ T$ is
        an isomorphism. Prove that $S$ is onto and $T$ is one-to-one. Give an
        example where $S$ is not one-to-one and $T$ is not onto.
    \item
        Prove that $T$ is onto if and only if $T(S)$ is spanning whenever $S$
        is spanning.
    \item
        Prove that $T$ is one-to-one if and only if $T(S)$ is linearly
        independent whenever $S$ is linearly independent.
    \item
        Prove that $T$ is an isomorphism if and only if $T(B)$ is a basis for
        any basis $B$.
    \item
        Suppose $\{u,v\}$ is a basis for $V$. Is $\{u-v,u+v\}$
        a basis for $V$?
    \item
        Suppose $\{u,v,w\}$ is a basis for $V$. Is $\{u-v,v-w,w-u\}$ a basis
        for $V$?
    \item
        (Definitely not on exam) Let $B$ be a basis for $\mathbf{R}$ as a
        $\mathbf{Q}$ vector space. Prove that $B$ is uncountable.
\end{enumerate}
\end{document}