(a) Let a, b € Rm be nonzero vectors. Prove that removing a's projection on the line of b from a leaves a vector perpendicular to b: bb™ br (arba) = 0 (0.0.1) Note: be aware of the dimension of each part to make sure you are canceling the correct part. (b) Let b₁,b2,..., b; be a list of nonzero mutually orthogonal vectors of Rm. Prove that removing each of a's projections to some b; from a leaves a vector perpendicular to each of Let b₁,b2, ..., bj : br (a - Σ i=1 T bibi -a) = 0, (for all k = 1,2, ..., j). b₂¹b₁ (0.0.2) This is almost trivial algebraically. Do you understand the geometric mean- ing of this, for example in R³(when j = 2)? Trying to draw it might help. Hint: You can discuss the relationship between k and i. Use (a) when they are the same, and use orthogonality when they are different.
(a) Let a, b € Rm be nonzero vectors. Prove that removing a's projection on the line of b from a leaves a vector perpendicular to b: bb™ br (arba) = 0 (0.0.1) Note: be aware of the dimension of each part to make sure you are canceling the correct part. (b) Let b₁,b2,..., b; be a list of nonzero mutually orthogonal vectors of Rm. Prove that removing each of a's projections to some b; from a leaves a vector perpendicular to each of Let b₁,b2, ..., bj : br (a - Σ i=1 T bibi -a) = 0, (for all k = 1,2, ..., j). b₂¹b₁ (0.0.2) This is almost trivial algebraically. Do you understand the geometric mean- ing of this, for example in R³(when j = 2)? Trying to draw it might help. Hint: You can discuss the relationship between k and i. Use (a) when they are the same, and use orthogonality when they are different.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Problem 1. (Gram - Schmidt procedure: proof)
(a) Let a, b E Rm be nonzero vectors. Prove that removing a's projection on the
line of b from a leaves a vector perpendicular to b:
bb™
bra)=
b (a-
(0.0.1)
Note: be aware of the dimension of each part to make sure you are canceling
the correct part.
bibi T
b₂¹b₂
= 0
(b) Let b₁,b2,..., b, be a list of nonzero mutually orthogonal vectors of Rm.
Prove that removing each of a's projections to some b; from a leaves a vector
perpendicular to each of Let b₁,b2, ..., bj :
b (a -
(0.0.2)
This is almost trivial algebraically. Do you understand the geometric mean-
ing of this, for example in R³(when j = 2)? Trying to draw it might help.
Hint: You can discuss the relationship between k and i. Use (a) when they
are the same, and use orthogonality when they are different.
-a)= 0, (for all k = 1, 2, ..., j).
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