For (a, B) E R?, define a function (-, -) : R² × R? → R by setting arr' + xy + Yx' + Byy' . 1. Find and sketch the set of (a, B) E R² for which (-, -) defines an inner product. 2. Is there is a choice of (a, B) e R? for which the vectors ). V2 are orthonormal with respect to (-, -)? 3. For this part of the question, you may assume that (-, -) is an inner product with a = orthogonality is defined in terms of this. 2 and B = 3, and that a. Starting from the canonical basis - (). () use Gram-Schmidt to find an orthonormal basis for R?. b. Compute the component of the vector (:) X = in the direction of e1. Hence write down the matrix that represents orthogonal projection onto the subspace spanned by e1.
For (a, B) E R?, define a function (-, -) : R² × R? → R by setting arr' + xy + Yx' + Byy' . 1. Find and sketch the set of (a, B) E R² for which (-, -) defines an inner product. 2. Is there is a choice of (a, B) e R? for which the vectors ). V2 are orthonormal with respect to (-, -)? 3. For this part of the question, you may assume that (-, -) is an inner product with a = orthogonality is defined in terms of this. 2 and B = 3, and that a. Starting from the canonical basis - (). () use Gram-Schmidt to find an orthonormal basis for R?. b. Compute the component of the vector (:) X = in the direction of e1. Hence write down the matrix that represents orthogonal projection onto the subspace spanned by e1.
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:For (a, B) E R?, define a function (-, -) : R² × R? → R by setting
arr' + xy +
Yx' + Byy' .
1. Find and sketch the set of (a, B) E R² for which (-, -) defines an inner product.
2. Is there is a choice of (a, B) e R? for which the vectors
).
V2
are orthonormal with respect to (-, -)?
3. For this part of the question, you may assume that (-, -) is an inner product with a =
orthogonality is defined in terms of this.
2 and B = 3, and that
a. Starting from the canonical basis
- ().
()
use Gram-Schmidt to find an orthonormal basis for R?.
b. Compute the component of the vector
(:)
X =
in the direction of e1. Hence write down the matrix that represents orthogonal projection onto the
subspace spanned by e1.
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