Linear Algebra

Section VI. Projection 257first represent the vector with respect to the basis. Then project the vector intothe span of each basis vector [⃗β 1 ] and [⃗β 2 ].(b) With this orthogonal basis for R 2( ( ) 1 1K = 〈 , 〉1)−1represent the same vector ⃗v with respect to the basis. Then project the vectorinto the span of each basis vector. Note that the coefficients in the representationand the projection are the same.(c) Let K = 〈⃗κ 1 , . . . ,⃗κ k 〉 be an orthogonal basis for some subspace of R n . Provethat 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 ).2.20 Bessel’s Inequality. Consider these orthonormal setsB 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 inB 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 thetwo 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 , andc 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 iscoefficient 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 andexpand the c’s.2.21 Prove or disprove: every vector in R n is in some orthogonal basis.2.22 Show that the columns of an n×n matrix form an orthonormal set if and onlyif the inverse of the matrix is its transpose. Produce such a matrix.2.23 Does the proof of Theorem 2.2 fail to consider the possibility that the set ofvectors is empty (i.e., that k = 0)?2.24 Theorem 2.7 describes a change of basis from any basis B = 〈⃗β 1 , . . . , ⃗β k 〉 toone that is orthogonal K = 〈⃗κ 1 , . . . ,⃗κ k 〉. Consider the change of basis matrixRep B,K (id).(a) Prove that the matrix Rep K,B (id) changing bases in the direction opposite tothat of the theorem has an upper triangular shape — all of its entries below themain 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.2.25 Complete the induction argument in the proof of Theorem 2.7.VI.3Projection Into a SubspaceThis subsection is optional. It also uses material from the optional earliersubsection on Combining Subspaces.