Dr. Marques Sophie
Office 519
Linear algebra
Spring Semester 2015
marques@cims.nyu.edu
Problem Set # 11
The following equations are considered over the reals numbers. All the answers should
be justified unless mentioned differently.
Problem 1 :
Let W be a subspace of Rn , and let W K be the set of all vectors orthogonal to W. Show
W K is a subspace of Rn .
Problem 2 :
Show that if x is in both W and W K , then x “ 0.
Problem 3 :
Determine which sets of vectors are orthogonal. If a set is is orthogonal, normalize the
vector to produce an orthonormal set.
1.
2.
ˆ
´0.6
0.8
˙ ˆ
˙
0.8
,
0.6
?
? ˛ ¨
˛ ¨
˛
0?
3{ ?10
1{ ?10
˝ 3{ 20 ‚, ˝ ´1{ 20 ‚, ˝ ´1{ 2 ‚
?
?
?
1{ 2
3{ 20
´1{ 20
¨
Problem 4 :
Given u ‰ 0 in Rn , let L “ Spantuu. Show that the mapping x ÞÑ projL pxq is a linear
transformation.
Problem 5 :
Express x as a linear combination of the u1 s.
ˆ
˙
ˆ ˙
ˆ
˙
2
6
9
u1 “
, u2 “
, and x “
´3
4
´7
Problem 6 :
Find the closest point to y in
¨
3
˚ 1
y“˚
˝ 5
1
the subspace W spanned by v1 and v2 .
˛
¨
˛
¨
˛
3
1
‹
˚
‹
˚
‹
‹ , v1 “ ˚ 1 ‹ and v2 “ ˚ ´1 ‹
‚
˝ ´1 ‚
˝ 1 ‚
1
´1
1
Problem 7 :
Find the best approximation to z by vectors of the form c1 v1 ` c2 v2 .
¨
˛
¨
˛
¨
˛
3
2
1
˚ ´7 ‹
˚
‹
˚
‹
‹ , v1 “ ˚ ´1 ‹ and v2 “ ˚ 1 ‹
z“˚
˝ 2 ‚
˝ ´3 ‚
˝ 0 ‚
1
´1
3
Problem 8 :
¨
˚
˚
A“˚
˚
˝
˛
1
2
5
´1 1 ´4 ‹
‹
´1 4 ´3 ‹
‹
1 ´4 7 ‚
1
2
1
1. Find an orthogonal basis for ColpAq.
2. Find a QR factorization of A.
2