Problem [1] 1. (1 mark) Consider the vector space of continuous functions C" ([-1, 1)). Give an example of a subspace. 2. (1 mark) Consider the space of square integrable functions ?([-1, 1]) with the usual inner product. Verify the Cauchy-Schwartz inequality for the functions f(x) = r and g(x) = ' 3. (2 marks) Determine all values of m so that the vectors form a basis of R3. { ( ) (:) (:) 4. (2 marks) If L : R3 - R', what are the dimensions of its associated matrix A? 5. (2 marks) Consider a matrix A E Max10(R). Given that dim(Ker A)=3, find dim(ImA). 6. (2 marks) Let do be an eigenvalue of a matrix A E Max.(C). Solve for a the equation ' + det(A - doIn) - 1 =0, where I, denotes the identity matrix in M, x.(C). 7. (2 marks) Let A E Max.(R) be a symmetric negative definite matrix. What is the sign of det.A