Suppose that T is a linear operator and x, y are eigenvectors corresponding to different eigenvalues i.e. Tx=λx and Tx=μy with λ and μ nonzero and x,y nonzero vectors. Then let us assume that x and y satisfy the equation x+cy=0 where c is a nonzero constant (i.e. let us assume that x and y are linearly dependent). By applying the operator T on both sides we get Tx+cTy=0 which is the same as λx+cμy=0 and since x=-cy we obtain that -cλy+cμy=0. Equivalently, cy(μ-λ)=0 which can not be true since c is a nonzero constant, y is a nonzero vector and the eigenvalues λ,μ are not equal. Therefore, the vectors x,y are linearly independent.
