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1 - every N x N matrix has N eigenvectors T/F ?

I think false because there would be infinitely many as any eigenvector can be multiplied by any scalar

2 - Operator R is diagonalizable where R = exp( i pi S

_{x}/hbar) where S

_{x}is the x spin operator

I think false because the x spin operator is not diagonal

3 - product of 2 unitary operators is a unitary operator T/F ?

i have no idea

4 - the exponential of a hermitian operator is a unitary operator T/F ?

I have no idea

The following questions all relate to finite dim complex vector space V

5 - the matrix representation of the identity operator is basis dependent T/F ?

i think false

6 - An orthogonal projector ( to a lower dim subspace) is neither injective nor surjective in V T/F ?

i think its not surjective , not sure about injective