- #1
Felafel
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Homework Statement
Given the endomorphism ϕ in ##\mathbb{E}^4## such that:
ϕ(x,y,z,t)=(4x-3z+3t, 4y-3x-3t,-z+t,z-t) find:
A)ker(ϕ)
B)Im(ϕ)
C)eigenvalues and multiplicities
D)eigenspaces
E)is ϕ self-adjoint or not? explain
The Attempt at a Solution
I get the associated matrix:
(4 0 -3 3)
(0 4 -3 -3)
(0 0 -1 1)
(0 0 1 -1)
but i can remove the last row, because it equals the third multiplied by -1
-solving AX=0 i have ker= L((0,0,1,1),(3, -3, 4, 4))
- reducing the columns i get Im= L((1, 0, 0),(0,1,0))
I'm not really sure this results are right, but what I wanted to ask is:
how do I compute the eigenvalues if the matrix is not square? is the rest of the exercise unsolvable?
thank you :)