@Misc{RANKNULL.MIZ,
  AUTHOR = {Alama, Jesse},
  TITLE = {The Rank+Nullity Theorem},
  DAY = {31},
  MONTH = {July},
  YEAR = {2007},
  ADDRESS1 = {Department of Philosophy\\Stanford University},
  SUMMARY = {The rank+nullity theorem states that, if $T$ is a
            linear transformation from a finite-dimensional
            vector space $V$ to a finite-dimensional vector
            space $W$, then $\dim(V) = \mathrm{rank}(T) +
            \mathrm{nullity}(T)$, where $\mathrm{rank}(T) =
            \dim(\mathrm{im}(T))$ and $\mathrm{nullity}(T) =
            \dim(\mathrm{ker}(T))$.  The proof treated here is
            standard; see, for example, \cite{lang-algebra}:
            take a basis $A$ of $\mathrm{ker}(T)$ and extend it
            to a basis $B$ of $V$, and then show that
            $\dim(\mathrm{im}(T)$ is equal to $|B - A|$, and that
            $T$ is one-to-one on $B - A$.},
  SECTION1 = {Preliminaries},
  SECTION2 = {Basic Facts of Linear Transformations},
  SECTION3 = {Some Lemmas on Linearly Independent Subsets,
           Linear Combinations, and Linear Transformations},
  SECTION4 = {The Rank+Nullity Theorem}}

@BOOK{lang-algebra,
  author = 	 {Lang, Serge},
  title = 	 {Algebra},
  publisher = 	 {Springer},
  year = 	 {2005},
  edition = 	 {3rd}}
