# proof of rank-nullity theorem

Let $T:V\to W$ be a linear mapping, with $V$
finite-dimensional^{}. We wish to show that

$$dimV=dimKerT+dimImgT$$ |

The images of a basis of $V$ will span $ImgT$, and hence $ImgT$ is finite-dimensional. Choose then a basis ${w}_{1},\mathrm{\dots},{w}_{n}$ of $ImgT$ and choose preimages ${v}_{1},\mathrm{\dots},{v}_{n}\in U$ such that

$${w}_{i}=T({v}_{i}),i=1\mathrm{\dots}n$$ |

Choose a basis ${u}_{1},\mathrm{\dots},{u}_{k}$ of $KerT$. The result will follow once we show that ${u}_{1},\mathrm{\dots},{u}_{k},{v}_{1},\mathrm{\dots},{v}_{n}$ is a basis of $V$.

Let $v\in V$ be given. Since $T(v)\in ImgT$, by definition, we can choose scalars ${b}_{1},\mathrm{\dots},{b}_{n}$ such that

$$T(v)={b}_{1}{w}_{1}+\mathrm{\dots}{b}_{n}{w}_{n}.$$ |

Linearity of $T$ now implies that $T({b}_{1}{v}_{1}+\mathrm{\dots}+{b}_{n}{v}_{n}-v)=0,$ and hence we can choose scalars ${a}_{1},\mathrm{\dots},{a}_{k}$ such that

$${b}_{1}{v}_{1}+\mathrm{\dots}+{b}_{n}{v}_{n}-v={a}_{1}{u}_{1}+\mathrm{\dots}{a}_{k}{u}_{k}.$$ |

Therefore ${u}_{1},\mathrm{\dots},{u}_{k},{v}_{1},\mathrm{\dots},{v}_{n}$ span $V$.

Next, let ${a}_{1},\mathrm{\dots},{a}_{k},{b}_{1},\mathrm{\dots},{b}_{n}$ be scalars such that

$${a}_{1}{u}_{1}+\mathrm{\dots}+{a}_{k}{u}_{k}+{b}_{1}{v}_{1}+\mathrm{\dots}+{b}_{n}{v}_{n}=0.$$ |

By applying $T$ to both sides of this equation it follows that

$${b}_{1}{w}_{1}+\mathrm{\dots}+{b}_{n}{w}_{n}=0,$$ |

and since ${w}_{1},\mathrm{\dots},{w}_{n}$ are linearly independent^{} that

$${b}_{1}={b}_{2}=\mathrm{\dots}={b}_{n}=0.$$ |

Consequently

$${a}_{1}{u}_{1}+\mathrm{\dots}+{a}_{k}{u}_{k}=0$$ |

as well, and since ${u}_{1},\mathrm{\dots},{u}_{k}$ are also assumed to be linearly independent we conclude that

$${a}_{1}={a}_{2}=\mathrm{\dots}={a}_{k}=0$$ |

also. Therefore ${u}_{1},\mathrm{\dots},{u}_{k},{v}_{1},\mathrm{\dots},{v}_{n}$ are linearly independent, and are therefore a basis. Q.E.D.

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

Canonical name | ProofOfRanknullityTheorem |

Date of creation | 2013-03-22 12:25:13 |

Last modified on | 2013-03-22 12:25:13 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 4 |

Author | rmilson (146) |

Entry type | Proof |

Classification | msc 15A03 |