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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?

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 1AEXP
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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?
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?
(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;
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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