section_path: Dot product and generalisations/Angles between subspaces
section_name: Angles between subspaces
paragraph_text: Angles between subspaces
  
  The definition of the angle between one-dimensional subspaces span (u) {\displaystyle \operatorname {span} (\mathbf {u})} and span (v) {\displaystyle \operatorname {span} (\mathbf {v})} given by
  
  | ⟨ u, v ⟩ | = | cos (θ) | ‖ u ‖ ‖ v ‖ {\displaystyle \left|\langle \mathbf {u},\mathbf {v} \rangle \right|=\left|\cos(\theta)\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|}
  
  in a Hilbert space can be extended to subspaces of finite dimensions. Given two subspaces U {\displaystyle {\mathcal {U}}}, W {\displaystyle {\mathcal {W}}} with dim (U):= k ≤ dim (W):= l {\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l}, this leads to a definition of k {\displaystyle k} angles called canonical or principal angles between subspaces.