Linear Algebra and Geometry. Written test of october 30, 2013.
1. Let B = {u, v, w} be an orthogonal basis of V
3such that k u k= 1, k v k= 2 and k w k= 3. (a) Compute the cosine of the angle between u − v + w and 2u − v − 2w.
(b) Find a basis of L(u − v + w)
⊥(express the vectors of the required basis as linear combinations of the vectors of B).
Solution. We start by recalling a well known (and easy) fact from the theory: if {u
1, · · · , u
n} is an orthogonal basis, u = P
ni=1
x
iu
iand v = P
ni=1
y
iu
ithen u · v =
n
X
i=1