Page 273 - 35Linear Algebra

14.7 Review Problems 273

(c) Suppose the vectors a, b and c, d are orthogonal. What can
you say about M in this case? (Hint: think about what M T is
equal to.)

2. Suppose S = {v 1 , . . . , v n } is an orthogonal (not orthonormal) basis
i
n
for R . Then we can write any vector v as v = P c v i for some
i
i
i
constants c . Find a formula for the constants c in terms of v and the
vectors in S.
Hint

3
3. Let u, v be linearly independent vectors in R , and P = span{u, v} be
the plane spanned by u and v.

⊥
(a) Is the vector v := v − u·v u in the plane P?
u·u
⊥
(b) What is the (cosine of the) angle between v and u?
⊥
(c) How can you find a third vector perpendicular to both u and v ?
3
(d) Construct an orthonormal basis for R from u and v.
(e) Test your abstract formulæ starting with

u = 1, 2, 0 and v = 0, 1, 1 .

Hint

4
4. Find an orthonormal basis for R which includes (1, 1, 1, 1) using the
following procedure:

(a) Pick a vector perpendicular to the vector

 
1
1
 
v 1 =  
1
 
1
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