Linear Algebra

Section VI. Projection 259
� � � � � � � � � � � �
1 2
0 −1
0 −1
(a) 〈 , 〉 (b) 〈 , 〉 (c) 〈 , 〉
1 1
1 3
1 0
Then turn those orthogonal bases into orthonormal bases.
� 2.10 Perform the Gram-Schmidt process on each of these bases for R 3 .
� � � � � � � � � � � �
2 1 0
1 0 2
(a) 〈 2 , 0 , 3 〉 (b) 〈 −1 , 1 , 3 〉
2 −1 1
0 0 1
Then turn those orthogonal bases into orthonormal bases.
� 2.11 Find an orthonormal basis for this subspace of R 3 : the plane x − y + z =0.
2.12 Find an orthonormal basis for this subspace of R 4 .
⎛ ⎞
x
⎜y
⎟
{ ⎝
z
⎠
w
� � x − y − z + w =0andx + z =0}
2.13 Show that any linearly independent subset of R n can be orthogonalized without
changing its span.
� 2.14 What happens if we apply the Gram-Schmidt process to a basis that is already
orthogonal?
2.15 Let 〈�κ1,...,�κk〉 be a set of mutually orthogonal vectors in R n .
(a) Prove that for any �v in the space, the vector �v−(proj [�κ1](�v )+···+proj [�vk](�v ))
is orthogonal to each of �κ1, ... , �κk.
(b) Illustrate the prior item in R 3 by using �e1 as �κ1, using�e2 as �κ2, and taking
�v to have components 1, 2, and 3.
(c) Show that proj [�κ1](�v )+···+proj [�vk](�v ) is the vector in the span of the set
of �κ’s that is closest to �v. Hint. To the illustration done for the prior part,
add a vector d1�κ1 + d2�κ2 and apply the Pythagorean Theorem to the resulting
triangle.
2.16 Find a vector in R 3 that is orthogonal to both of these.
� � � �
1 2
5 2
−1 0
� 2.17 One advantage of orthogonal bases is that they simplify finding the representation
of a vector with respect to that basis.
(a) For this vector and this non-orthogonal basis for R 2
� � � � � �
2
1 1
�v = B = 〈 , 〉
3
1 0
first represent the vector with respect to the basis. Then project the vector into
the span of each basis vector [ � β1] and[ � β2].
(b) With this orthogonal basis for R 2
� � � �
1 1
K = 〈 , 〉
1 −1
represent the same vector �v with respect to the basis. Then project the vector
into the span of each basis vector. Note that the coefficients in the representation
and the projection are the same.
(c) Let K = 〈�κ1,... ,�κk〉 be an orthogonal basis for some subspace of R n .Prove
that for any �v in the subspace, the i-th component of the representation RepK(�v )
is the scalar coefficient (�v �κi)/(�κi �κi) from proj [�κi] (�v ).