\documentclass{amsart}


\title{Matrix theory homework 5} 

\begin{document}

\maketitle


\begin{enumerate}


\item    The {\em rowspace} of an $m \times n$ matrix $A$ is the set of linear combinations
         of the rows of $A$. Prove that the rowspace of $A$ is the set of row vectors of the
         form $y A$. Prove that if $E$ is an invertible $m \times m$ matrix then the 
         rowspace of $A$ is equal to the rowspace of $E A$. 
     
\item     Let $A$ be a square matrix. Prove that if $A$ is invertible then the transpose $A^T$ is invertible.
          
\item     Let $A$ be an invertible square matrix. For $n > 0$ we already defined $A^n$ to be the product of
          $n$ copies of $A$. We let $A^0 = 1_{n \times n}$ and  for $n < 0$ we define $A^{-n} = (A^{-1})^n$. 
          Prove that for all integers $m, n$ we have $A^{m + n} = A^m A^n$ and $(A^m)^n = A^{m n}$. 

\item     Let $A$ be an $m \times m$ matrix, and for $t > 0$ let $X_t$ be the columnspace of $A^t$. 
          Prove that  $X_{t +1} \subseteq X_t$ for all $t > 0$. Prove that there is a number $T >0$ such that
          $X_t = X_T$ for all $t \ge T$.  

\end{enumerate} 

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