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(d) Use result from (c) to sketch 2r² + 2xy + 2y2 = 3 based on the coordinate system formed by u and as the axes. Hint: You might want to revise on sketching ellipse.

(d) Use result from (c) to sketch 2r² + 2xy + 2y2 = 3 based on the coordinate system formed by u and as the axes. Hint: You might want to revise on sketching ellipse.

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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Hi, could you please answer part d for me 

3. Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² +2bxy+ay² = c with b #0 and c> 0. a (öd). b Let V(x, y) = ax² + 2bry + ay2 and denote A = Note that: V(x, y) = (x y) A ^ (*) = ((*), ^ (*)) where (,): R2 x R2 → R is the standard inner (dot) product on R². (a) Determine all eigenvalues of A, and their corresponding eigenspaces. (b) Construct an orthonormal basis ü, ü of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector ER2 as a linear combination of u and . (;) (e) Use result from (b) to transform V(x, y) = ((*). 4 ()) into V(u, v) au²+ Bu² for some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A.
Transcribed Image Text:3. Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. We will focus on the equation in below form: ax² +2bxy+ay² = c with b #0 and c> 0. a (öd). b Let V(x, y) = ax² + 2bry + ay2 and denote A = Note that: V(x, y) = (x y) A ^ (*) = ((*), ^ (*)) where (,): R2 x R2 → R is the standard inner (dot) product on R². (a) Determine all eigenvalues of A, and their corresponding eigenspaces. (b) Construct an orthonormal basis ü, ü of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector ER2 as a linear combination of u and . (;) (e) Use result from (b) to transform V(x, y) = ((*). 4 ()) into V(u, v) au²+ Bu² for some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A.
(d) Use result from (c) to sketch 22² + 2xy + 2y²: = 3 based on the coordinate system formed by and as the axes. Hint: You might want to revise on sketching ellipse. 16 a mustory board game players move between locations
Transcribed Image Text:(d) Use result from (c) to sketch 22² + 2xy + 2y²: = 3 based on the coordinate system formed by and as the axes. Hint: You might want to revise on sketching ellipse. 16 a mustory board game players move between locations
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