1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices...
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...
1. Let A be the 2x2 matrix in M_{2x2}(C), whose first row is
(0,1) and second...
1. Let A be the 2x2 matrix in M_{2x2}(C), whose first row is
(0,1) and second row is (-1,0).
(a) Show that A is normal.
(b) Find (complex) eigenvalues of A.
(c) Find an orthogonal basis for C^2, which consists of
eigenvectors of A.
(d) Find an orthonormal basis for C^2, which consists of
eigenvectors of A.
U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the...
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
Let M=[[116,−48],[−48,44]]. Notice that 20 is an eigenvalue of
M. Let U be an orthogonal matrix...
Let M=[[116,−48],[−48,44]]. Notice that 20 is an eigenvalue of
M. Let U be an orthogonal matrix such that (U^−1)(M)(U) is
diagonal, the first column of U has positive entries, and det(U)=1.
Find (√20)⋅U.
The matrix B that has
(1,2,3,1)
(0,1,-1,2)
as rows. Try computing the determinant of B*(B^T) without...
The matrix B that has
(1,2,3,1)
(0,1,-1,2)
as rows. Try computing the determinant of B*(B^T) without computing
B*B^T directly, by using the Cauchy-Binnet theorem.
Assume that a and b are two continuous injective maps with
a(0)=(-1,0), a(1)=(1,0), b(0)=(0,-1) and b(1)=(0,1)....
Assume that a and b are two continuous injective maps with
a(0)=(-1,0), a(1)=(1,0), b(0)=(0,-1) and b(1)=(0,1). Show that the
curves a and b must intersect
1) If u and v are orthogonal unit vectors, under what condition
au+bv is orthogonal to...
1) If u and v are orthogonal unit vectors, under what condition
au+bv is orthogonal to cu+dv (where a, b, c, d are scalars)? What
are the lengths of those vectors (express them using a, b, c,
d)?
2) Given two vectors u and v that are not orthogonal, prove that
w=‖u‖2v−uuT v is orthogonal to u, where ‖x‖ is the L^2 norm of
x.