---- > [!definition] Definition. ([[Euler characteristic of a complex of vector spaces]]) > Let $k$ be a [[field]] and consider a [[chain complex of modules|complex]] of $k$-[[vector space|vector spaces]] and [[linear map|linear maps]]: $V_{\bullet}: 0 \to V_{N} \xrightarrow{\ \ \alpha_{N}\ \ } V_{N-1} \xrightarrow{\alpha_{N-1}} \cdots \xrightarrow{\ \ \ \alpha_{2}\ \ \ } V_{1} \xrightarrow{ \ \ \ \alpha_{1 \ \ \ }} V_{0} \to 0.$ The **Euler characteristic** of $V_{\bullet}$ is the integer $\chi(V_{\bullet}):= \sum_{i} (-1)^{i} \ \text{dim}(V_{i}),$ where $\text{dim}(V_{i})$ denotes the [[dimension]] of $V_{i}$. ^definition > [!equivalence] > - [[computing the Euler characteristic of a vector space complex using homology]] ^equivalence ---- #### ---- #### 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 > ```