quality:correction, structural:summarization, linguistic:translation, information:classification
Finding projection with an inner product

Let V {\displaystyle V} be a vector space (in this case a plane) spanned by orthogonal vectors u 1, u 2, …, u p {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\dots,\mathbf {u} _{p}}. Let y {\displaystyle y} be a vector. One can define a projection of y {\displaystyle \mathbf {y} } onto V {\displaystyle V} as proj V y = y ⋅ u i u i ⋅ u i u i {\displaystyle \operatorname {proj} _{V}\mathbf {y} ={\frac {\mathbf {y} \cdot \mathbf {u} ^{i}}{\mathbf {u} ^{i}\cdot \mathbf {u} ^{i}}}\mathbf {u} ^{i}} where repeated indices are summed over (Einstein sum notation). The vector y {\displaystyle \mathbf {y} } can be written as an orthogonal sum such that y = proj V y + z {\displaystyle \mathbf {y} =\operatorname {proj} _{V}\mathbf {y} +\mathbf {z} }. Note that proj V y {\displaystyle \operatorname {proj} _{V}\mathbf {y} } is sometimes denoted as y ^ {\displaystyle {\hat {\mathbf {y} }}}. There is a theorem in linear algebra that states that this z {\displaystyle \mathbf {z} } is the smallest distance (the orthogonal distance) from y {\displaystyle \mathbf {y} } to V {\displaystyle V} and is commonly used in areas such as machine learning.