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\begin{document}

\title[101 Daily Homework \#1]{Math 101: Graduate Linear Algebra \\ Daily Homework \#1}
\date{Assigned Monday, 11 September 2017; due Wednesday, 13 September 2017}

\maketitle

\begin{problab}{1.1}
Let $f\colon X \to Y$ be a function.  Given another function $g\colon Y \to Z$, we can compose to get $g\circ f \colon X \to Z$ defined by $(g \circ f)(x)=g(f(x))$.  Sometimes we will have more elaborate diagrams:
\[
\xymatrix{
X \ar[r]^f \ar[dr]_h & Y \ar[d]^g \\
& Z 
} \]
We say a diagram like the above is \defi{commutative} if we start from one set and travel to any other, we get the same answer regardless of the path chosen: in the above example, this reads $h(x)=g(f(x))$ for all $x \in X$.

We say that $f$ \defi{factors through} a map $g\colon X \to Z$ if there exists a map $h\colon Z \to Y$ such that $f(x)=h(g(x))$ for all $x \in X$, and we say $f$ \defi{factors uniquely through} $g$ if the map $h$ is unique.

Define the relation $\sim$ on $X$ by $x \sim x'$ if $f(x)=f(x')$.  
\begin{enumalph}
\item Show that $\sim$ is an equivalence relation.
\item Show that $f$ factors uniquely through the projection $\pi\colon X \to X/\!\sim$.  If $f$ is surjective, show further that the map $(X/\!\sim) \to Y$ is bijective.  Draw the corresponding commutative diagram.
\item Now let $F$ be a field, let $V,W$ be $F$-vector spaces, and let $\phi\colon V \to W$ be an $F$-linear map.  Show that $\phi$ factors uniquely through the quotient $V \to V/\ker \phi$.  If $\phi$ is surjective, what more can you say?
\end{enumalph}
\end{problab}

\end{document}