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\centerline{\Large Math 675 Homework 1}
\centerline{Due 8/29/2018}
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Responses must be typed, with a two-page maximum and LaTeX highly recommended. Grade will be based on style (2 points) and the best three problems (2 points each), but all five are highly recommended.

\begin{enumerate}[1.]
\item Let $\{v_1, \ldots, v_m\}\subset \C^m$ be nonzero vectors such that $(v_i, v_j)=0$ when $i\neq j$. Show that the set $\{v_1, \ldots, v_m\}$ is linearly independent.
\item Let $\{v_1, \ldots, v_m\}\subset \C^m$ be a set of linearly independent vectors, and suppose that $w\in \C^m\setminus \operatorname{span}\{v_1, \ldots, v_m\}$.  Prove that the set $\{v_1, \ldots, v_m, w\}$ is linearly independent.
\item Let $U, V\subset \C^m$ be two subspaces. Prove that
$$\dim(U+V)+\dim(U\cap V)=\dim U + \dim V.$$
\item Let $A$ be an $m\times n$ matrix. Show that the null space and range of $A$ satisfy
$$\dim N(A)+\dim R(A)=n.$$
\item Let $A$ be an $m\times n$ matrix. Show that 
$$R(A)^\perp =N(A^*).$$
\end{enumerate}




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