Properties:: [[basic properties of the alt function]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- > [!definition] Definition. ([[the alternization function]]) > We define the [[linear map]] $\text{Alt}:$ [[vector space of k-tensors]] $\to$ [[vector space of alternating k-tensors on a vector space]] by $\text{Alt}(f)(\v v_{1},\dots, \v v_{k}) = \sum_{\sigma \in S_{k}}^{} \text{sgn}(\sigma)\cdot f\big(\v v_{\sigma(1)},\dots, \v v_{\sigma(k)}\big).$ > [!justification]- > We must show that $\text{Alt}$ is [[linear map]] and that, indeed, $f \in \text{Mult}^{k}(V) \implies \text{Alt}(f) \in \Lambda^{k}(V)$.![[CleanShot 2023-01-27 at 21.32.57.jpg]] Or can equivalently define on pure tensors as $\text{Alt}( \theta_{1} \otimes \dots \otimes \theta_{k} )=\sum_{\sigma \in S_{k}} \text{sgn}(\sigma) \cdot \theta_{\sigma(1)} \otimes \dots \otimes \theta_{\sigma(k)}$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```