3. Eigenvalues and eigenvectors are useful in sketching complicated graph of conic sections. Wewill focus on the equation in below form:ax² +2bxy+ay² = c with b #0 and c> 0.a(öd).bLet 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 afixed 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 forsome 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 formedby and as the axes.Hint: You might want to revise on sketching ellipse.16amustory board game players move between locations

As per norms, we will be answering the first three subparts. If you need an answer to others, then…

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 (öö). b Let V(x, y) = ax² +2bry + ay² and denote A = Note that: v (2, 1) = (2 v) A (*) = ((*), 4 (*)) А 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, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector (+) ER2 as a linear combination of u and . (c) Use result from (b) to transform (z,y) = ((*). 4 ()) A some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into V(u, v) = au² + Bv² for

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 (öö). b Let V(x, y) = ax² +2bry + ay² and denote A = Note that: v (2, 1) = (2 v) A () = ((), 4 ()) А 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, 7 of R2, formed by eigenvectors of A. Then, express a fixed but unknown vector (+) ER2 as a linear combination of u and . (c) Use result from (b) to transform (z,y) = ((). 4 ()) A some u(x, y) and v(x, y), with a, ß ER being eigenvalues of A. into V(u, v) = au² + Bv² for

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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Question

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

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