Linear Algebra, 2020a

Section VI. Projection 285
(c) Let K = 〈⃗κ 1 ,...,⃗κ k 〉 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 Rep K (⃗v )
is the scalar coefficient (⃗v • ⃗κ i )/(⃗κ i
• ⃗κ i ) from proj [⃗κi ](⃗v ).
(d) Prove that ⃗v = proj [⃗κ1 ](⃗v )+···+ proj [⃗κk ](⃗v ).
2.23 Bessel’s Inequality. Consider these orthonormal sets
B 1 = {⃗e 1 } B 2 = {⃗e 1 ,⃗e 2 } B 3 = {⃗e 1 ,⃗e 2 ,⃗e 3 } B 4 = {⃗e 1 ,⃗e 2 ,⃗e 3 ,⃗e 4 }
along with the vector ⃗v ∈ R 4 whose components are 4, 3, 2, and 1.
(a) Find the coefficient c 1 for the projection of ⃗v into the span of the vector in
B 1 . Check that ‖⃗v ‖ 2 |c 1 | 2 .
(b) Find the coefficients c 1 and c 2 for the projection of ⃗v into the spans of the
two vectors in B 2 . Check that ‖⃗v ‖ 2 |c 1 | 2 + |c 2 | 2 .
(c) Find c 1 , c 2 , and c 3 associated with the vectors in B 3 , and c 1 , c 2 , c 3 , and
c 4 for the vectors in B 4 . Check that ‖⃗v ‖ 2 |c 1 | 2 + ···+ |c 3 | 2 and that ‖⃗v ‖ 2
|c 1 | 2 + ···+ |c 4 | 2 .
Show that this holds in general: where {⃗κ 1 ,...,⃗κ k } is an orthonormal set and c i is
coefficient of the projection of a vector ⃗v from the space then ‖⃗v ‖ 2 |c 1 | 2 +···+|c k | 2 .
Hint. One way is to look at the inequality 0 ‖⃗v −(c 1 ⃗κ 1 + ···+ c k ⃗κ k )‖ 2 and
expand the c’s.
2.24 Prove or disprove: every vector in R n is in some orthogonal basis.
2.25 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.
2.26 Does the proof of Theorem 2.2 fail to consider the possibility that the set of
vectors is empty (i.e., that k = 0)?
2.27 Theorem 2.7 describes a change of basis from any basis B = 〈⃗β 1 ,...,⃗β k 〉 to
one that is orthogonal K = 〈⃗κ 1 ,...,⃗κ k 〉. 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.
2.28 Complete the induction argument in the proof of Theorem 2.7.
VI.3
Projection Into a Subspace
This subsection uses material from the optional earlier subsection on Combining
Subspaces.
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 will follow this decomposition idea.