Show that: ∑ x i e i = 0 and also show that ∑ y ^ i e i = 0. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that ∑ e i = 0.

Question

Show that:   ∑      x    i        e    i    =  0 and also show that   ∑              y      ^        i        e    i    =  0. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that   ∑      e    i    =  0.

Kitamiliseakekw · Accepted Answer

Here&#039;s one way of viewing it. We want to write      [                                        y            1                                                ⋮                                                  y            n                                ]    =      α    ^        [                            1                                      ⋮                                      1                      ]    +      β    ^        [                                        x            1                                                ⋮                                                  x            n                                ]    +      [                                                                ε              ^                        1                                                ⋮                                                  ε            n                                ]  and choose the values of       α    ^   and       β    ^   that minimze               ε      ^        1    2    +  ⋯  +              ε      ^        n    2  . The sum of the first two terms on the right is   [              y      ^        1    ,  …  ,              y      ^        n        ]    T  .That means the point       [                                                                y              ^                        1                                                ⋮                                                  y            n                                ]    =      α    ^        [                            1                                      ⋮                                      1                      ]    +      β    ^        [                                        x            1                                                ⋮                                                  x            n                                ]   is closer to       [                                        y            1                                                ⋮                                                  y            n                                ]   in ordinary Euclidean distance than is any other point in the plane spanned by       [                            1                                      ⋮                                      1                      ]   and       [                                        x            1                                                ⋮                                                  x            n                                ]  . The point in a plane that is closet to   y is the point you get by dropping a perpendicular from   y to the plane. That means       ε    ^    =        y    ^    −  y is perpendicular to the two columns that span the plane, and thus perpendicular to every linear combination of them, such as         y    ^  . &quot;Perpendicular&quot; means the dot-product is zero. Q.E.D.The vector of fitted values         y    ^    =      [                                                                y              ^                        1                                                ⋮                                                                          y              ^                        n                                ]   is the orthogonal projection of the vector   y  =      [                                        y            1                                                ⋮                                                  y            n                                ]   onto the column space of the design matrix   X  =      [                            1                                      x            1                                                ⋮                          ⋮                                      1                                      x            n                                ]  .The orthogonal projection is a linear transformation whose matrix is the &quot;hat matrix&quot; is   H  =  X  (      X    T    X      )          −      1            X    T  , an   n  ×  n matrix of rank   2. Observe that if   w is orthogonal to that column space then   X  w  =  0 so   H  w  =  0, and if   w is in the column space, then   w  =  X  u for some   u  ∈        R    2  , and so   H  w  =  w.It follows that       ε    ^    =  y  −        y    ^    =  (  I  −  H  )  y is orthogonal to the column space. Since         y    ^   is in the column space,       ε    ^   is orthogonal to         y    ^  .

Show that: sum x_ ie_i=0 and also show that sum y_i e_i=0. Now I do believe that being able to solve the first sum will make the solution to the second sum more clear. So far I have proved that sum e_i=0.

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