Determine the orthogonal projection of x 4 4 3 onto the sub

Determine the orthogonal projection of x = (4, 4, 3) onto the subspace of R^3 spanned by the two vectors u = (4, 1, -2) and y = (1, -2, 1). Notice that . Check your work by verifying independently the two properties that the projection should have.

Solution

A=[U,V]= X= U V 4 4 1 4 1 -2 3 -2 1 A\'= A TRANSPOSE = 4 1 -2 1 -2 1 A\'*A= 21 0 0 6 [A\'*A] INVERSE = 0.0476 0 0 0.1667 PROJECTION MATRIX = A*[{A\'*A} INVERSE]*A\' = 4 1 0.0476 0 4 1 -2 1 -2     * 0 0.1667     * 1 -2 1 -2 1 PROJECTION MATRIX = A*[{A\'*A} INVERSE]*A\' = 4 1 0.1905 0.0476 -0.095 1 -2     * 0.1667 -0.333 0.1667 -2 1 PROJECTION MATRIX = A*[{A\'*A} INVERSE]*A\' = P = 0.9286 -0.143 -0.214 -0.143 0.7143 -0.429 -0.214 -0.429 0.3571 ORTHOGONAL PROJECTION OF X ON TO A = P*X= 0.9286 -0.143 -0.214 4 2.5 -0.143 0.7143 -0.429    * 4     = 1 -0.214 -0.429 0.3571 3 -1.5