Determine if the following are subspaces ProveCounterexample

Determine if the following are subspaces (Prove/Counterexample):

a) The set of 2x2 matrices with det(A) = 0 inside M₂₂(R).

c) The set of functions with f \'(x) = 1 inside F over R.

d)The set of vectors (x, y, 0) inside R³ over R.

Solution

Post multiple question to get the remaining answer

a) The set is not closed under scalar addition

Reason: Considering two matrices A = [1 0; 0 0] and B = [0 0; 0 1], the matrix A and B both have the determinant equal to zero, but the resulting matrix A+B is equal to [1 0; 0 1], the determinant of A+B is not equal to 0 and it is equal to 1, hence the matrix is not closed under scalar addition

Therefore, the following is not a subspace