Give examples of Two different inner products for P4. A 2*2 matrix that is diagonalizable over the complex numbers but not over the real numbers.

Give examples of

  1. Two different inner products for P4.
  2. A 2*2 matrix that is diagonalizable over the complex numbers but not over the real numbers.
  3. A 4*4 matrix that is not diagonal but is diagonalizable.
  4. A 3*3 matrix that is not diagonalizable.
  5. A subspace of P3 of dimension 2 and containing the vector X2-1.
  6. A non invertible square matrix.

Appreciated for any explaining.