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
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: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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa923b6f-81dd-482c-8885-6de6bc295751%2F536d3bb5-7bca-420e-973a-7f1d07e3f116%2Fju65cz7_processed.jpeg&w=3840&q=75)
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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