Exercise 1 (5 points):Leta, bbe real numbers.Consider the following system ofequations:X+Y+ 2Z=a2X+ 2Y+ 3Z=b3X+ 3Y+ 4Z=a+b(1) Determine all possible values ofa, bfor which the above system has a solution.When the system has a solution, describe all solutions in terms ofaandb.(2) Are there any real numbersa, bfor which the system of equations above hasexactly one solution?Solution:
Exercise 2 (5 points):(1) Give an example of 2⇥2 matricesCandDsuch thatCD6=DC.(2) LetA= (a00a) withaa real number. Show that ifBis any 2⇥2 matrix, thenAB=BA.(3) Are there any other 2⇥2 matricesAhaving the property thatAB=BAfor all2⇥2 matricesB? Hint: Start by considering matrices likeB= (1 00 0).Solution:
Exercise 3 (5 points):LetVbe a real vector space.Suppose thatv1, v2, v3, v4arevectors inVwhich are linearly independent. Show that the vectorsv1,v1+v2,v1+v2+v3,v1+v2+v3+v4are also linearly independent.