rank-nullity theorem

Let $V$ and $W$ be vector spaces over the same field.If $\varphi :V\to W$ is a linear mapping, then

$$\dim V=\dim (\ker \varphi )+\dim (\mathrm{im}\varphi ).$$

In other words, the dimension of $V$is equal to the sum (http://planetmath.org/CardinalArithmetic)of the rank (http://planetmath.org/RankLinearMapping) and nullity of $\varphi$.

Note that if $U$ is a subspace of $V$, then this(applied to the canonical mapping $V\to V/U$) says that

$$\dim V=\dim U+\dim (V/U),$$

that is,

$$\dim V=\dim U+\mathrm{codim}U,$$

where $\mathrm{codim}$ denotes codimension.

An alternative way of stating the rank-nullity theorem isby saying that if

$$0\to U\to V\to W\to 0$$

is a short exact sequence of vector spaces, then

$$\dim (V)=\dim (U)+\dim (W).$$

In fact, if

$$0\to V_1\to \mathrm{\cdots }\to V_n\to 0$$

is an exact sequence of vector spaces, then

$$\sum _{i=1}^{\lfloor n/2\rfloor }{V}_{2i}=\sum _{i=1}^{\lceil n/2\rceil }{V}_{2i-1},$$

that is, the sum of the dimensions of even-numbered termsis the same as the sum of the dimensions of the odd-numbered terms.