----- - Let [[inner product space]] be an [[inner product space|inner product space]]. > [!proposition] Proposition. ([[columns linearly independent iff gram matrix is invertible]]) > A set of [[vector]]s $u_{1},\dots,u_{N} \in V$ is [[linearly independent]] if and only if the [[Gram matrix]] defined by $G_{ij}:=\langle u_{i}, u_{j} \rangle$ is [[inverse matrix|nonsingular]]. > [!proof]- Proof. ([[columns linearly independent iff gram matrix is invertible]]) > Suppose $u_{1},\dots,u_{N}$ is [[linearly independent]], so that $U := [u_{1} \ \dots \ u_{N}]$ has full [[column rank]]. Since $\ker U'U = \ker U$, $\ker U'U = \{ 0 \}$. So $U'U$ is invertible. > > Conversely suppose $U'U$ is [[inverse matrix|invertible]]. Then $\ker U'U = \{ 0 \}=\ker U$. So $U$ has full column rank. So $u_{1},\dots,u_{N}$ is [[linearly independent]]. > > > ----- #### ---- #### 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 > ```