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
Related questions
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe31e29b-6809-420d-ac12-25e2458c2da4%2Ff01cd193-5c79-4b04-9515-f3e77606599f%2Fkqcwapk_processed.jpeg&w=3840&q=75)
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](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbe31e29b-6809-420d-ac12-25e2458c2da4%2Ff01cd193-5c79-4b04-9515-f3e77606599f%2Fvb53run_processed.jpeg&w=3840&q=75)
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
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 5 steps
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Follow-up Question
is it possible to have this written out? thanks
Solution
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)