rank-nullity theoremRankNullityTheorem2002-02-19 18:56:162002-02-22 10:58:33Theorem\usepackage{amsmath}
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\newtheorem{proposition}{Proposition}The sum of the rank and the nullity of a linear mapping gives
the dimension of the mapping's domain. More precisely, let
$T:V\rightarrow W$ be a linear mapping. If $V$ is a
finite-dimensional, then
$$\dim V = \dim \mathop{\mathrm{Ker}} T + \dim \mathop{\mathrm{Img}}
T.$$
The intuitive content of the Rank-Nullity theorem is the principle that
\begin{quote}\em
Every independent linear constraint takes away one degree of freedom.
\end{quote}
The rank is just the number of independent linear constraints on $v\in
V$ imposed
by the equation
$$T(v)=0.$$
The dimension of $V$ is the number of unconstrained degrees of freedom.
The nullity is the degrees of freedom in the resulting space of
solutions.
To put it yet another way:
\begin{quote} \em
The number of variables {\bf minus} the number of independent linear
constraints {\bf equals}
the number of linearly independent solutions.
\end{quote}