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

Find the 3 * 3 matrix A corresponding to orthogonal projection
onto the solution
space of the system below.
2x + 3y + z = 0;
x - 3y - z = 0:
Your solution should contain the following information: (a) The
eigenvector(s) of
A that is (are) contained in the solution space; (b) The
eigenvector(s) of A that
is (are) perpendicular to the solution space; (c) The corresponding
eigenvalues for
those eigenvectors.

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

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