3. (a) Use change-of-basis techniques to construct a 2 x 2 matrix A that satisfies the following eigenvector equations: A 3 4 (²)=³(²), A(-¹2) = 5 (-²2). Please write your matrix A relative to the standard basis. (b) Now suppose that {v₁, v2} is a basis for R2 and that A₁, A2 are scalars. Describe a general formula to construct a 2 x 2 matrix B such that Bu; = Aivi for i=1, 2, where B is written with respect to the standard basis. How many possible matrices B are there that satisfy these equations?
3. (a) Use change-of-basis techniques to construct a 2 x 2 matrix A that satisfies the following eigenvector equations: A 3 4 (²)=³(²), A(-¹2) = 5 (-²2). Please write your matrix A relative to the standard basis. (b) Now suppose that {v₁, v2} is a basis for R2 and that A₁, A2 are scalars. Describe a general formula to construct a 2 x 2 matrix B such that Bu; = Aivi for i=1, 2, where B is written with respect to the standard basis. How many possible matrices B are there that satisfy these equations?
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:3. (a) Use change-of-basis techniques to construct a 2 × 2 matrix A that satisfies the following
eigenvector equations:
2
2
* () = ³ (1)
A
"
A
= 5
Please write your matrix A relative to the standard basis.
=
(b) Now suppose that {v₁, v₂} is a basis for R² and that №₁, №2 are scalars. Describe a general
formula to construct a 2 × 2 matrix B such that Bvį Aivi for i 1,2, where B is written
with respect to the standard basis. How many possible matrices B are there that satisfy these
equations?
=
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