# Linear algebra problem

1. Apr 23, 2013

### Felafel

1. The problem statement, all variables and given/known data
Write a selfadjoint endomorphism $f : E^3 → E^3$ such that $ker(f ) = L((1, 2, 1))$ and $λ_1 = 1, λ_2 = 2$ are eigenvalues of f

3. The attempt at a solution

I know $λ_3=0$ because ́$ker(f ) ≠ {(0, 0, 0)}$ and $(ker(f ))^⊥ = (V0 )^⊥ = V1 ⊕ V2$ due to the definition of selfadjoint.
Then, my book gives the solution:
$(ker(f ))^⊥ = (L((1, 2, 1))^⊥ = {(α, β, −α − 2β) | α, β ∈ R} = L((1, 0, −1), (a, b, c))$
but i don't understand where did it get that (α, β, −α − 2β) from.
Could you please help me? thanks in advance :)

2. Apr 23, 2013

### Dick

(α, β, −α − 2β) is the subspace orthogonal to (1,2,1). You know that the eigenvectors of a self adjoint operator corresponding to different eigenvectors are orthogonal, yes? So you'll have to pick the other two eigenvectors from that space.

Last edited: Apr 23, 2013
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted