1. The problem statement, all variables and given/known data Prove without induction that Multiplicity of an eigen value , k = dim[ Null(T - k I)^( dim V) ] 2. Relevant equations [(T - k I)^dim V ] v =0 [Thoughts] i understand that normal eigen vectors with same eigen values may not be linearly independent. [(T - k I)^dim V ] v =0 then, the fact that k = dim[ Null(T - k I)^dim V ] somehow gives an intuition that in this case, the Eigen vectors with the same Eigen value k are linearly independent ? This is confusing to me. 3. The attempt at a solution If i can know, that for [(T - k I)^dim V ] v =0 , the solutions are linearly independent, then the desired result can be proved. OR if i prove that the solutions to the above equation are eigen vectors which form a basis, then i have the solution. What could be a direction ?