3. Define a function to be a set of N points, with the m'th point being the value of the function on the m'th point in the set. Then a basis would be the standard N-component unit vectors (1 in one position, 0 otherwise).

4. Are you working over C or over R? The 2x2 complex matrices have 8 degrees of freedom if over R, and 4 if over C - without your additional condition. The condition A transpose = A complex conjugate imposes the condition that the diagonal elements are real, and the off diagonal elements are related - just write out both of the matrices like you did in the problem and see what the condition implies in terms of the matrix elements.