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Let W be the set of all vectors = E R such that xı – 2x2 + 3r3 = 0. (Geometrically, W is a plane passing through the origin in R³; algebraically, W is a two-dimensional subspace of R³.) Find an orthogonal basis {v1, 02} for W. (Hint: You can find a matrix A such that W = null(A).)

Let W be the set of all vectors = E R such that xı – 2x2 + 3r3 = 0. (Geometrically, W is a plane passing through the origin in R³; algebraically, W is a two-dimensional subspace of R³.) Find an orthogonal basis {v1, 02} for W. (Hint: You can find a matrix A such that W = null(A).)

Advanced Engineering Mathematics
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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Let W be the set of all vectors = r2 E R° such that x1 – 2x2 + 3x3 = 0. (Geometrically, W is a plane passing through the origin in R³; algebraically, W is a two-dimensional subspace of R³.) Find an orthogonal basis {ī1, 02} for W. (Hint: You can find a matrix A such that W = null(A).)
Transcribed Image Text:Let W be the set of all vectors = r2 E R° such that x1 – 2x2 + 3x3 = 0. (Geometrically, W is a plane passing through the origin in R³; algebraically, W is a two-dimensional subspace of R³.) Find an orthogonal basis {ī1, 02} for W. (Hint: You can find a matrix A such that W = null(A).)
Andwer Key W = null([1 -2 3]) (this matrix is already in RREF), so by our usual method for finding bases for null spaces, a basis for W is {0F} This basis is not yet orthogonal; Gram-Schmidt gives us the desired orthogonal basis, -3/5] 6/5 where if you like you can scale the second vector by 5 to clear fractions (the result will still be an orthogonal basis).
Transcribed Image Text:Andwer Key W = null([1 -2 3]) (this matrix is already in RREF), so by our usual method for finding bases for null spaces, a basis for W is {0F} This basis is not yet orthogonal; Gram-Schmidt gives us the desired orthogonal basis, -3/5] 6/5 where if you like you can scale the second vector by 5 to clear fractions (the result will still be an orthogonal basis).
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