# Section3Range, nullspace, and rank (NM)¶ permalink

More from Lecture 1 of Trefethen and Bau.

# Subsection3.1Exercises

##### Exercise3.1

Prove that $A \in \Cmm$ has full rank if and only if it maps no two distinct vectors to the same vector. That is, show that if $x$ and $y$ are vectors, with $x \neq y\text{,}$ then

\begin{equation*} Ax \neq Ay \iff \Rank(A)=m. \end{equation*}
##### Exercise3.2

Suppose you know the rank of the $m\times m$ matrices $A$ and $B\text{.}$ What, if anything, can you say about the rank of $C=AB\text{?}$ Suppose $A$ or $B$ have full rank, what can you say about the rank of $C\text{?}$