For the first one: you need to consider the three axioms of the sub-space which are namely a) the zero vector being in the space, b) closure in the sub-space for scalar multiplication and addition.
The basis for a sub-space does not have to be the same as that for the whole space.
For number 2, you should firstly note that the transformation is square and secondly that you have non-zero eigenvalues if the determinant is non-zero.
When does the determinant with respect to the derivative be non-zero? (Hint: Think about when you have a minimum, maximum in a multivariable scenario).