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\title{Math 490 Homework 2\\ Due: February 9}

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\maketitle

Your homework should be written in \LaTeX. 

\begin{enumerate}
\item In class, we computed the homology of a connected graph. Generalize this computation to an arbitrary (finite) graph. 

\item Let $F$ be a field and let $\phi$ be an $m \times n$ matrix with entries in $F$. 

Given a non-negative integer $k$, a $k \times k$ minor of $\phi$ is the determinant of a $k \times k$ submatrix of $\phi$. Prove that the rank of $\phi$ is $r$ if and only if all $(r+1) \times (r+1)$ minors of $\phi$ are $0$, and some $r \times r$ minor is nonzero. 

(The determinant of a $0 \times 0$ matrix is defined to be $1$.)

\item Consider the following chain complex\footnote{This computes the homology of the real projective plane ${\bf RP}^2$ but that fact isn't needed.}:
\[
\bZ^2 \xrightarrow{\begin{pmatrix} -1 & 1 \\ 1 & -1 \\ 1 & 1 \end{pmatrix}} \bZ^3 \xrightarrow{\begin{pmatrix} -1 & -1 & 0 \\ 1 & 1 & 0 \end{pmatrix}} \bZ^2
\]
\begin{enumerate}
\item Show that ${\rm H}_2 \cong 0$, ${\rm H}_1 \cong \bZ/2$, and ${\rm H}_0 \cong \bZ$.

\item Use problem 2 to compute the ranks of the two matrices with $\bZ$ replaced by the field of rational numbers ${\bf Q}$ and also with the finite field $\bZ/p$ where $p$ is a prime. 

\item Use (b) to compute the dimensions of the homology groups when $\bZ$ is replaced by one of the fields ${\bf Q}$ or $\bZ/p$.
\end{enumerate}
\end{enumerate}

\newpage

Hints: 

\begin{enumerate}
\item Different connected components of a graph don't interact with each other when applying the boundary map.
\item The rank of a matrix is the dimension of its column space, and also the dimension of its row space. A square matrix has determinant $0$ if and only if its columns are linearly dependent if and only if its rows are linearly independent.
\item 
For (a), to calculate ${\rm H}_1$, first find a spanning set for the kernel of the right map and a spanning set for the image of the left map. Then show there are exactly 2 different cosets when quotienting the kernel by the image.

\end{enumerate}

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