linear-guest

G.13 Orthonormal Bases and Complements 427
(d) Construct an orthonormal basis for R 3 from u and v.
If you did part (c) you can probably find 3 orthogonal vectors to make
a orthogonal basis. All you need to do to turn this into an orthonormal
basis is make these into unit vectors.
(e) Test your abstract formulae starting with
u = ( 1 2 0 ) and v = ( 0 1 1 ) .
Try it out, and if you get stuck try drawing a sketch of the vectors you
have.
Hint for Review Problem 10
This video shows you a way to solve problem 10 that’s different to the method
described in the Lecture. The first thing is to think of
⎛
1 0
⎞
2
M = ⎝−1 2 0⎠
−1 2 2
as a set of 3 vectors
⎛ ⎞
⎛ ⎞
⎛ ⎞
0
0
2
v 1 = ⎝−1⎠ , v 2 = ⎝ 2⎠ , v 3 = ⎝0⎠ .
−1
−2
2
Then you need to remember that we are searching for a decomposition
M = QR
where Q is an orthogonal matrix. Thus the upper triangular matrix R = Q T M
and Q T Q = I. Moreover, orthogonal matrices perform rotations. To see this
compare the inner product u v = u T v of vectors u and v with that of Qu and
Qv:
(Qu) (Qv) = (Qu) T (Qv) = u T Q T Qv = u T v = u v .
Since the dot product doesn’t change, we learn that Q does not change angles
or lengths of vectors.
Now, here’s an interesting procedure: rotate v 1 , v 2 and v 3 such that v 1 is
along the x-axis, v 2 is in the xy-plane. Then if you put these in a matrix you
get something of the form
⎛ ⎞
a b c
⎝0 d e⎠
0 0 f
which is exactly what we want for R!
427