(Solved): Given the linear transformation l:R3R3,l(x,y,z)=(2x+y+3z,x+3y+z,x) find a basis B of eigenve ...

Given the linear transformation $$l:{\mathbb{R}}^3?{\mathbb{R}}^3,l(x,y,z)=(2x+y+3z,x+3y+z,x)$$ find a basis $$B$$ of eigenvectors of $$l$$, such that the matrix $$\mathbf{D}$$ associated to $$l$$ with respect to $$B$$ is diagonal. Verify that $${\mathbf{P}}^{?1}\mathbf{AP}=\mathbf{D}$$, where $$\mathbf{P}$$ is the transition matrix from basis $$B$$ to the canonical one, and $$\mathbf{A}$$ is the matrix associated to $$l$$ with respect to the canonical basis.