Page 427 - 35Linear Algebra

G.13 Orthonormal Bases and Complements 427

3
(d) Construct an orthonormal basis for R 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

T
where Q is an orthogonal matrix. Thus the upper triangular matrix R = Q M
T
and Q Q = I. Moreover, orthogonal matrices perform rotations. To see this
T
compare the inner product u v = u v of vectors u and v with that of Qu and
Qv:
T
T
T
T
(Qu) (Qv) = (Qu) (Qv) = u Q Qv = u 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!

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