Answer to Question #81384 in Linear Algebra for Ankush Sarkar

Question

which of the following statements are true and which are false ? justify your answer with a short proof or a counterexample. i) r^2 has infinitely many non zero, proper vector subspaces. ii) if t:v -> w is one-one linear transformation between two finite dimensional vector spaces v and w then t is invertible. iii) if a^k = 0 for a square matrix a, then all the eigen values of a are non zero. iv) every unitary operator is invertible. v) every system of homogeneous linear equations has a non zero solution.