Download Answers to Even-Numbered Homework Problems, Section 6.2 20

Answers to Even-Numbered Homework Problems, Section 6.2
20. Let

u=
− 23
1
3
2
3



and v = 
1
3
2
3
0

.
Since u · v = 0, {u, v} is an orthogonal set. However, kuk2 = u · u = 1 and kvk2 = v · v =
5
6= 1, so {u, v} is not an orthonormal set. The vector v can be normalized, with
9
ṽ =

3
1
v= √ v=
kvk
5
√1
5
√2
5
0

,
so that {u, ṽ} is an orthonormal set. (Note that u is already a unit vector.)
26. A set of n nonzero orthogonal vectors must be linearly independent by Theorem 4, so if
such a sets spans W , it is a basis for W . Since W is therefore an n-dimensional subspace
of Rn , it must be equal to Rn itself.
28. If U is an n × n orthogonal matrix, then In = U U −1 = U U T . Since U is the transpose
of U T , that is, since (U T )T = U , Theorem 6 applied to U T says that U T has orthogonal
columns. In particular, the columns of U T are linearly independent and hence form a basis
for Rn , by the Invertible Matrix Theorem. That is, the rows of U form an orthonormal
basis for Rn .