Question

1. Find the orthogonal projection of the matrix [[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form lambda?I.  

[[4.5,0][0,4.5]]  [[5.5,0][0,5.5]]  [[4,0][0,4]]  [[3.5,0][0,3.5]]  [[5,0][0,5]]  [[1.5,0][0,1.5]]

2. Find the orthogonal projection of the matrix [[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace 0.  

[[-1,3][3,1]]  [[1.5,1][1,-1.5]]  [[0,4][4,0]]  [[3,3.5][3.5,-3]]  [[0,1.5][1.5,0]]  [[-2,1.5][1.5,2]]  [[0.5,4.5][4.5,-0.5]]  [[-1,6][6,1]]  [[0,3.5][3.5,0]]  [[-1.5,3.5][3.5,1.5]]

3. Find the orthogonal projection of the matrix [[1,5][1,2]] onto the space of anti-symmetric 2x2 matrices.  

[[0,-1] [1,0]]  [[0,2] [-2,0]]  [[0,-1.5] [1.5,0]]  [[0,2.5] [-2.5,0]]  [[0,0] [0,0]]  [[0,-0.5] [0.5,0]]  [[0,1] [-1,0]] [[0,1.5] [-1.5,0]]  [[0,-2.5] [2.5,0]]  [[0,0.5] [-0.5,0]]

4. Let p be the orthogonal projection of u=[40,-9,91]^T onto the column space of matrix M= [[4,7,3][-4,1,3]]^T . Find vector X so that p=M?X.

[3,2]^T [4,5]^T [5,6]^T [1,4]^T [7,3]^T [4,3]^T [5,4]^T [2,6]^T [3,5]^T [7,4]^T

Answer 1