Question 3 (12 marks)
Consider the four points O(0; 0; 0), A(1; 1; 0), B(0; 1; 1) and C(1; 2; 4) in R3.
(a) Find a vector equation of the line L containing B and C.
(b) Find a Cartesian equation for the plane II1 containing O, A and B.
(c) Find a Cartesian equation for the plane II2 containing A, B and C.
(d) Find the cosine of angle between the planes II1 and II2 .
(e) Find the volume of the parallelepiped having OA, OB and OC as edges
Question 4 (10 marks)
In each part of this question, determine whether W is a subspace of the real vector space V .
For each part, give a complete proof using the subspace theorem, or a specic counterexample
to show that some subspace property fails.
Question 6 (10 marks)
Let B = fb1; b2; b3g be an ordered basis for a real vector space V , and suppose that
c1 = b1; c2 = b1 b2; c3 = b1 3b2 + b3
(a) Write down the coordinate vectors [c1]B, [c2]B and [c3]B.
(b) Show that C = fc1; c2; c3g is a basis for V .
(c) Find the transition matrix PB;C (that converts coordinates with respect to C to coordi-
nates with respect to B).
(d) Find the transition matrix PC;B (that converts coordinates with respect to B to coordi-
nates with respect to C).
(e) Using your answer to part (d), nd [2b + 3b b ] .