A First Course in Linear Algebra, 2017a

254 R n
Exercise 4.11.7 For U an orthogonal matrix, explain why ‖U⃗x‖ = ‖⃗x‖ for any vector ⃗x. Next explain why
if U is an n × n matrix with the property that ‖U⃗x‖ = ‖⃗x‖ for all vectors, ⃗x, then U must be orthogonal.
Thus the orthogonal matrices are exactly those which preserve length.
Exercise 4.11.8 Suppose U is an orthogonal n × n matrix. Explain why rank(U)=n.
Exercise 4.11.9 Fill in the missing entries to make the matrix orthogonal.
⎡
⎤
⎢
⎣
−1 √
2
−1 √
6
1 √3
1√
2
_ _
_
√
6
3
_
⎥
⎦ .
Exercise 4.11.10 Fill in the missing entries to make the matrix orthogonal.
⎡ √
2 2
√ ⎤
1
3 2 6 2
⎢
⎣
2
3
_ _
_ 0 _
⎥
⎦
Exercise 4.11.11 Fill in the missing entries to make the matrix orthogonal.
⎡
1
3
− 2 ⎤
√ 5
_
⎢ 2
⎣ 3
0 _ ⎥
√ ⎦
4
_ _ 15 5
Exercise 4.11.12 Find an orthonormal basis for the span of each of the following sets of vectors.
(a)
(b)
(c)
⎡
⎣
⎡
⎣
⎡
⎣
3
−4
0
3
0
−4
3
0
−4
⎤
⎡
⎦, ⎣
⎤
⎡
⎦, ⎣
⎤
⎡
⎦, ⎣
7
−1
0
11
0
2
5
0
10
⎤
⎡
⎦, ⎣
⎤
⎡
⎦, ⎣
⎤
⎡
⎦, ⎣
1
7
1
1
1
7
⎤
⎦
⎤
⎦
−7
1
1
⎤
⎦