Suppose V is a vector space and T is a linear operator on V

Suppose V is a vector space and T is a linear operator on V. Prove that if u and v are eigenvectors for T associated with distinct eigenvalues, then {u,v} is a linearly independent set.

Solution

Given that V is a vector space and T is a linear operator on V.

c1 and c2 are two distinct eigen values of A, the matrix assoicated with T.

Since c1 not equal to c2,

Eigen vectors v1 willnot be equal to v2

Even by elementary operations eigen vector v1 cannot be made as a linear combination of v2

Hence v1 and v2 are linearly independent.