Suppose V is an n-dimensional space, (,) is an inner product and {b₁,b} is a basis for V. We say the basis (b₁,b} is or- thonormal (with respect to (-.-)) if i (bi, bj) = 0 if i #j; ii (b₁, b;) = 1 for all i Le. the length of b;'s are all one. Answer the following: (a) Check whether the standard basis in R" with the Euclidean norm (or dot product) is an orthonormal basis. (b) Check whether the following is a basis for R² {0.4]} Is it an orthonormal basis (with the Euclidean norm)? (c) Multiply each vector above by. Is this now an orthonormal basis? (d) Suppose we have a basis (b₁,,b} where condition (a) holds Le. (bi, bj) = 0 for ij. How to generate an orthonormal basis from it?
Suppose V is an n-dimensional space, (,) is an inner product and {b₁,b} is a basis for V. We say the basis (b₁,b} is or- thonormal (with respect to (-.-)) if i (bi, bj) = 0 if i #j; ii (b₁, b;) = 1 for all i Le. the length of b;'s are all one. Answer the following: (a) Check whether the standard basis in R" with the Euclidean norm (or dot product) is an orthonormal basis. (b) Check whether the following is a basis for R² {0.4]} Is it an orthonormal basis (with the Euclidean norm)? (c) Multiply each vector above by. Is this now an orthonormal basis? (d) Suppose we have a basis (b₁,,b} where condition (a) holds Le. (bi, bj) = 0 for ij. How to generate an orthonormal basis from it?
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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![1. Suppose V is an n-dimensional space, (,) is an inner product and
{b₁,bn} is a basis for V. We say the basis (b₁,b₂} is or-
thonormal (with respect to (,)) if
i. (bi, bj) = 0 if i #j;
ii. (b₁, b₁) = 1 for all i i.e. the length of b;'s are all one.
Answer the following:
(a) Check whether the standard basis in R" with the Euclidean norm
(or dot product) is an orthonormal basis.
(b) Check whether the following is a basis for R²
{0.4]}.
Is it an orthonormal basis (with the Euclidean norm)?
(c) Multiply each vector above by. Is this now an orthonormal
basis?
(d) Suppose we have a basis {b₁,b₂} where condition (a) holds
i.e. (b₁,bj) = 0 for ij. How to generate an orthonormal basis
from it?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb5247a99-177c-47ac-b687-b87a8e03f9b5%2Fa7362123-a860-4729-8164-af0b7adc80b9%2Fubradvb_processed.png&w=3840&q=75)
Transcribed Image Text:1. Suppose V is an n-dimensional space, (,) is an inner product and
{b₁,bn} is a basis for V. We say the basis (b₁,b₂} is or-
thonormal (with respect to (,)) if
i. (bi, bj) = 0 if i #j;
ii. (b₁, b₁) = 1 for all i i.e. the length of b;'s are all one.
Answer the following:
(a) Check whether the standard basis in R" with the Euclidean norm
(or dot product) is an orthonormal basis.
(b) Check whether the following is a basis for R²
{0.4]}.
Is it an orthonormal basis (with the Euclidean norm)?
(c) Multiply each vector above by. Is this now an orthonormal
basis?
(d) Suppose we have a basis {b₁,b₂} where condition (a) holds
i.e. (b₁,bj) = 0 for ij. How to generate an orthonormal basis
from it?
Transcribed Image Text:(e) The following process is called Gram-Schmidt process for orthogo-
nalization: Assume {b₁,,b} is a given basis. Then construct
bi=b₁,
b = b₂-projb; (b₂),
b = b - projы (bs) - projь (b),
b=b4-projb; (b4) – proj; (b4) - proj; (b4).
E
b=b₁-proj; (bn),
i=1
We claim {bi,...,b;} is a basis that all the distinct elements are
orthogonal to each other.
Check whether b; and b; are orthogonal to each other.
• Consider the Euclidean norm and apply the Gram-Schmidt
B
• Consider the Euclidean norm and apply the Gram-Schmidt
process to 0
and 1 in R³.
(f) Generate an orthonormal basis in each case above.
process to
and in R³;
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