Euclidean vector spaces 1. Given the vectors a = [1, 2, 0, 2] and b = [-2, 0, 1, 1], find: (i) a + 2b (ii) The unit vector b (iii) A vector in the same direction as b but has the same length of a 2. Given the points A(2, 4,3, -1, 1) and B(3, 1, 1, 0, -2) in IR, find the distance be- tween the points A and B. 3. For the vectors a = [4, 1, -2, 2] and b = [1, 0,3, 2] determine the vector projection of a on b. 4. Find the angle between the hyperplanes 21-12-213+IA = -1 and 1+312-14 = 2. Vector subspaces 5. For each of following sets of vectors, determine whether or not it is a subspace of R', giving reasons for your answer. I (i) V= y N (ini) w = NEN ER yzERLinear combinations 6. Let v1= and va = E R. Show that w = is a linear combination of v1, v2 and v3. 7. Let v. - 1 ."-3]-ex-[ 3 E R. Show that w = -3 can not be written as a linear combination of v1, v2 and va. Linear dependence / independence 8. For each of the following sets of vectors, decide whether they are l.i. or I.d