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*Problem 3.4 Suppose you start out with a basis (lei), le2), ..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le2), ..., le)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (ele₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from le3) its projections along le₁) and le₂): les) - (ejles) lej) - (e₂lez)|e₂). This is orthogonal to le) and le₂); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)i + (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28)ĵ+ (0)k.

*Problem 3.4 Suppose you start out with a basis (lei), le2), ..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le2), ..., le)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (ele₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from le3) its projections along le₁) and le₂): les) - (ejles) lej) - (e₂lez)|e₂). This is orthogonal to le) and le₂); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)i + (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28)ĵ+ (0)k.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 18E
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*Problem 3.4 Suppose you start out with a basis (le1), le₂),..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le₂),..., len)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le₁) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (eile₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from e3) its projections along le) and le₂): les) - (ejles) lej) - (e₂|e3)|e2). This is orthogonal to le) and le2); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)î+ (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28) ĵ + (0)k.
Transcribed Image Text:*Problem 3.4 Suppose you start out with a basis (le1), le₂),..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le₂),..., len)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le₁) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (eile₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from e3) its projections along le) and le₂): les) - (ejles) lej) - (e₂|e3)|e2). This is orthogonal to le) and le2); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)î+ (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28) ĵ + (0)k.
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