# another proof of rank-nullity theorem

Let $\varphi :V\to W$ be a linear transformation from vector spaces^{} $V$ to $W$. Recall that the rank of $\varphi $ is the dimension^{} of the image of $\varphi $ and the nullity^{} of $\varphi $ is the dimension of the kernel of $\varphi $.

###### Proposition 1.

$\mathrm{dim}(V)=\mathrm{rank}(\varphi )+\mathrm{nullity}(\varphi )$.

###### Proof.

Let $K=\mathrm{ker}(\varphi )$. $K$ is a subspace^{} of $V$ so it has a unique algebraic complement $L$ such that $V=K\oplus L$. It is evident that

$$\mathrm{dim}(V)=\mathrm{dim}(K)+\mathrm{dim}(L)$$ |

since $K$ and $L$ have disjoint bases and the union of their bases is a basis for $V$.

Define ${\varphi}^{\prime}:L\to \varphi (V)$ by restriction^{} of $\varphi $ to the subspace $L$. ${\varphi}^{\prime}$ is obviously a linear transformation. If ${\varphi}^{\prime}(v)=0$, then $\varphi (v)={\varphi}^{\prime}(v)=0$ so that $v\in K$. Since $v\in L$ as well, we have $v\in K\cap L=\{0\}$, or $v=0$. This means that ${\varphi}^{\prime}$ is one-to-one. Next, pick any $w\in \varphi (V)$. So there is some $v\in V$ with $\varphi (v)=w$. Write $v=x+y$ with $x\in K$ and $y\in L$. So ${\varphi}^{\prime}(y)=\varphi (y)=0+\varphi (y)=\varphi (x)+\varphi (y)=\varphi (v)=w$, and therefore ${\varphi}^{\prime}$ is onto. This means that $L$ is isomorphic^{} to $\varphi (V)$, which is equivalent^{} to saying that $\mathrm{dim}(L)=\mathrm{dim}(\varphi (V))=\mathrm{rank}(\varphi )$. Finally, we have

$$\mathrm{dim}(V)=\mathrm{dim}(K)+\mathrm{dim}(L)=\mathrm{nullity}(\varphi )+\mathrm{rank}(\varphi ).$$ |

∎

Remark. The dimension of $V$ is not assumed to be finite in this proof. For another approach (where finite dimensionality of $V$ is assumed), please see this entry (http://planetmath.org/ProofOfRankNullityTheorem).

Title | another proof of rank-nullity theorem |
---|---|

Canonical name | AnotherProofOfRanknullityTheorem |

Date of creation | 2013-03-22 18:06:14 |

Last modified on | 2013-03-22 18:06:14 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 4 |

Author | CWoo (3771) |

Entry type | Proof |

Classification | msc 15A03 |

Related topic | ProofOfRankNullityTheorem |