---- > [!definition] Definition. ([[linearly independent vector fields]]) > Let $M$ be a [[smooth manifold]] of dimension $n$. An ordered $k$-tuple $X^{(1)},\dots,X^{(k)}$ of [[vector field|vector fields]] on some subset $A \subset M$ is said to be **linearly independent** if $(X^{(1)} |_{p},\dots,X^{(k)}|_{p})$ is a [[linearly independent]] $k$-tuple in the [[tangent space at a point of a smooth manifold|tangent space]] $T_{p}M$ at each $p \in A$. > We say $X^{(1)},\dots,X^{(k)}$ **spans the tangent bundle** if $(X^{(1)} |_{p},\dots,X^{(k)}|_{p})$ [[submodule generated by a subset|spans]] $T_{p}M$ for each $p \in A$. ^definition ---- #### ---- #### 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 > ```