# some facts about injective and surjective linear maps

Let $k$ be a field and $V,W$ be vector spaces^{} over $k$.

Proposition^{}. Let $f:V\to W$ be an injective^{} linear map. Then there exists a (surjective^{}) linear map $g:W\to V$ such that $g\circ f={\mathrm{id}}_{V}$.

Proof. Of course $\mathrm{Im}(f)$ is a subspace^{} of $W$ so $f:V\to \mathrm{Im}(f)$ is a linear isomorphism. Let ${({e}_{i})}_{i\in I}$ be a basis of $\mathrm{Im}(f)$ and ${({e}_{j})}_{j\in J}$ be its completion to the basis of $W$, i.e. ${({e}_{i})}_{i\in I\cup J}$ is a basis of $W$. Define $g:W\to V$ on the basis as follows:

$$g({e}_{i})={f}^{-1}({e}_{i}),\text{if}i\in I;$$ |

$$g({e}_{j})=0,\text{if}j\in J.$$ |

We will show that $g\circ f={\mathrm{id}}_{V}$.

Let $v\in V$. Then

$$f(v)=\sum _{i\in I}{\alpha}_{i}{e}_{i},$$ |

where ${\alpha}_{i}\in k$ (note that the indexing set is $I$). Thus we have

$$(g\circ f)(v)=g(\sum _{i\in I}{\alpha}_{i}{e}_{i})=\sum _{i\in I}{\alpha}_{i}g({e}_{i})=\sum _{i\in I}{\alpha}_{i}{f}^{-1}({e}_{i})=$$ |

$$={f}^{-1}(\sum _{i\in I}{\alpha}_{i}{e}_{i})={f}^{-1}(f(v))=v.$$ |

It is clear that the equality $g\circ f={\mathrm{id}}_{V}$ implies that $g$ is surjective. $\mathrm{\square}$

Proposition. Let $g:W\to V$ be a surjective linear map. Then there exists a (injective) linear map $f:V\to W$ such that $g\circ f={\mathrm{id}}_{V}$.

Proof. Let ${({e}_{i})}_{i\in I}$ be a basis of $V$. Since $g$ is onto, then for any $i\in I$ there exist ${w}_{i}\in W$ such that $g({w}_{i})={e}_{i}$.
Now define $f:V\to W$ by the formula^{}

$$f({e}_{i})={w}_{i}.$$ |

It is clear that $g\circ f={\mathrm{id}}_{V}$, which implies that $f$ is injective. $\mathrm{\square}$

If we combine these two propositions, we have the following corollary:

Corollary. There exists an injective linear map $f:V\to W$ if and only if there exists a surjective linear map $g:W\to V$.

Title | some facts about injective and surjective linear maps |
---|---|

Canonical name | SomeFactsAboutInjectiveAndSurjectiveLinearMaps |

Date of creation | 2013-03-22 18:32:22 |

Last modified on | 2013-03-22 18:32:22 |

Owner | joking (16130) |

Last modified by | joking (16130) |

Numerical id | 6 |

Author | joking (16130) |

Entry type | Derivation^{} |

Classification | msc 15A04 |