Linear Algebra

260 Chapter 3. Maps Between Spaces
(d) Prove that �v =proj [�κ1](�v )+···+proj [�κk](�v ).
2.18 Bessel’s Inequality. Consider these orthonormal sets
B1 = {�e1} B2 = {�e1,�e2} B3 = {�e1,�e2,�e3} B4 = {�e1,�e2,�e3,�e4}
along with the vector �v ∈ R 4 whose components are 4, 3, 2, and 1.
(a) Find the coefficient c1 for the projection of �v into the span of the vector in
B1. Checkthat��v � 2 ≥|c1| 2 .
(b) Find the coefficients c1 and c2 for the projection of �v into the spans of the
two vectors in B2. Checkthat��v � 2 ≥|c1| 2 + |c2| 2 .
(c) Find c1, c2, andc3 associated with the vectors in B3, andc1, c2, c3, andc4
for the vectors in B4. Check that ��v � 2 ≥|c1| 2 + ···+ |c3| 2 and that ��v � 2 ≥
|c1| 2 + ···+ |c4| 2 .
Show that this holds in general: where {�κ1,... ,�κk} is an orthonormal set and ci is
coefficient of the projection of a vector �v from the space then ��v � 2 ≥|c1| 2 + ···+
|ck| 2 . Hint. One way is to look at the inequality 0 ≤��v − (c1�κ1 + ···+ ck�κk)� 2
and expand the c’s.
2.19 Prove or disprove: every vector in R n is in some orthogonal basis.
2.20 Show that the columns of an n×n matrix form an orthonormal set if and only
if the inverse of the matrix is its transpose. Produce such a matrix.
2.21 Does the proof of Theorem 2.2 fail to consider the possibility that the set of
vectors is empty (i.e., that k =0)?
2.22 Theorem 2.7 describes a change of basis from any basis B = 〈 � β1,... , � βk〉 to
one that is orthogonal K = 〈�κ1,... ,�κk〉. Consider the change of basis matrix
Rep B,K(id).
(a) Prove that the matrix Rep K,B(id) changing bases in the direction opposite
to that of the theorem has an upper triangular shape—all of its entries below
the main diagonal are zeros.
(b) Prove that the inverse of an upper triangular matrix is also upper triangular
(if the matrix is invertible, that is). This shows that the matrix Rep B,K(id)
changing bases in the direction described in the theorem is upper triangular.
2.23 Complete the induction argument in the proof of Theorem 2.7.
3.VI.3 Projection Into a Subspace
This subsection, like the others in this section, is optional. It also requires
material from the optional earlier subsection on Direct Sums.
The prior subsections project a vector into a line by decomposing it into two
parts: the part in the line proj [�s ](�v ) and the rest �v − proj [�s ](�v ). To generalize
projection to arbitrary subspaces, we follow this idea.
3.1 Definition For any direct sum V = M ⊕ N and any �v ∈ V ,theprojection
of �v into M along N is
where �v = �m + �n with �m ∈ M, �n ∈ N.
proj M,N(�v )=�m