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csc2iffy
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Homework Statement
Define L:R3-->R3 by L(x,y,z)=(y-z,x+z,-x+y).
A. Show that L is self-adjoint using the standard orthonormal basis B of R3.
B. Diagonalize L and find and orthogonal basis B of R3 of eigenvectors of L and the diagonal matrix.
C. Considering only the eigenvalues of L, determine if L is an isomorphism.
D. Find L(1,0,0) using the diagonal matrix of L.
Homework Equations
L(x,y,z)=(y-z,x+z,-x+y)
Matrix of L with respect to orthonormal basis:
0 1 -1
1 0 1
-1 1 0
Diagonal matrix of L:
1 0 0
0 1 0
0 0 -2
The Attempt at a Solution
I already answered A and B. For C, I said L is not an isomorphism because of repeated eigenvalues. For D, I am not sure how to find the linear transformation using only the diagonal matrix, but I know the answer is (0,1,-1).