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\textbf{Exercises on the four fundamental subspaces}
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\begin{problem}(3.6 \#11.  {\em Introduction to Linear Algebra:} Strang) $A$ is an $m$ by $n$ matrix of rank $r$. Suppose there are right sides $\vb b$ for which $A\mathbf{x = b}$ has \textit{no solution}.
\begin{enumerate}[a)]
\item What are all the inequalities ($<$ or $\leq$) that must be true between $m, n,$ and $r$?

\item How do you know that $A^T \mathbf{y =0}$ has solutions other than $\mathbf{y=0}$?
\end{enumerate}
\end{problem}

\solution{
\begin{enumerate}[a)]
\item The rank of a matrix is always less than or equal to the number of rows and columns, so $r \leq m$ and $r \leq n$. The second statement tells us that the column space is not all of $\R^n$, so $r < m.$

\item These solutions make up the left nullspace, which has dimension $m -r > 0$ (that is, there are nonzero vectors in it).

\end{enumerate}
}

\begin{problem}(3.6 \#24.) $A^T \mathbf{y =d}$ is solvable when $\vb d$ is in which of the four subspaces? The solution $\vb y$ is unique when the $\rule{20mm}{0.4pt}$ contains only the zero vector. 
\end{problem}

\solution{ It is solvable when $\vb d$ is in the row space, which consists of all vectors $A^T \vb y.$ The solution $\vb y$ is unique when the \textbf{left nullspace} contains only the zero vector.
}

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