Problem 2. Let A & Fnxn be a fixed matrix. Define the following set IAC F[x] as follows:IA = {ƒ|ƒ(A) = 0nxn}1. Show that IA is an ideal.2. Recall we proved the following theorem (as a corollary of the discussion) in the lastclass.Theorem 1. If IC F[x] is an ideal, a unique monic polynomial f I such that forany geI f\g.In the context of Problem 2.1, such unique polynomial fa that generates the ideal I iscalled the minimal polynomial for the matrix A. Find the minimal polynomial for thefollowing matrix: A =00100

Answer:1. Yes, IA is an ideal due to the closure properties mentioned above.2. The minimal…

Answered: Problem 2. Let A & Fnxn be a fixed…

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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3. A quadratic function in n variables is any function defined on R" which can be expressed in the form f(x) = a + b·x + x• Ax, where a € R, be R", and A is an n × n-symmetric matrix. (a) Show that the function f(x) defined on R² by f(x₁, x₂) = (x₁ - x₂)² + (x₁ + 2x₂ + 1)² − 8x₁x₂ is a quadratic function of two variables by finding the appropriate a € R, b = R², and the 2 x 2-symmetric matrix A. (b) Compute the gradient Vf(x) and the Hessian Hf(x) of the quadratic function in (a) and express these quantities in terms of the a € R, b = R², and the 2 x 2-symmetric matrix A computed in (a). (c) Show that a quadratic function f(x) of n variables is convex if and only if the corresponding n × n-symmetric matrix A is positive semidefinite, and is strictly convex if A is positive definite. (d) If f(x) is a quadratic function of n variables such that the corresponding matrix A is positive definite, show that 0 2Ax + b has a unique solution and that this solution is the strict global minimizer…
Q. 8.(a) Let {e} be an orthonormal set in a Hilbert space H, then Σ|(x,e₁) | ² ≤||x||² for each vector x in H.
Problem 4: Consider the set of polynomials {2+ t, 1 – 5t}. Let p¡(t) = 2 + t and P2(t) = 1 – 5t. Let B = {2+1, 1 – 5t}. (a) Show that the set B is a basis for P1. Show your work (define a basis if you use a coordinate mapping). (b) If [v (1)], find v(t). B (c) Find [1- 51 (d) Find -2+ 43t] B (e) Find 0 (t) where 0 (t) = 0 + Ot. B
- 5. A linear operator T: V → V is said to be nilpotent if Th 0 for some positive integer k. Similarly, a square matrix A is said to be nilpotent if Ak = 0 for some positive integer k. = (a) Let T: V → V be a linear operator on a finite-dimensional complex inner product space V. Show that if T is normal and nilpotent, then T = 0. (Hint: spectral theorem for normal operators.) (b) Use (a) to show that if a square matrix A is normal and nilpotent, then A = 0. (Hint: apply (a) to T = LA: Cn Cn.) →
4. Let {f1, f2, f3} be the dual basis of {e₁, €2, €3} for R³, where e₁ = (1, 1, 1), €₂=(1,1,−1), e3=(1,-1,-1). Find f₁(x), f₂(x), f(x), where X = = (1, 0, 0).
5.PS-16 Question Help A shoe store uses a 60% markup for all of the shoes it sells. What would be the selling price of a pair of shoes that has a wholesale cost of $55? The selling price would be $ (Round to the nearest cent as needed.) Enter your answer in the answer box and then click Check Answer. All narte chowing Review progress Question 9. of 9 pe here to search
10. [H] Let A be a 2 x2 real matrix such that AX that A al for some a € R. = XA for all 2 x 2 real matrices X. Show
Find the basis for both Range(T) and Ker(T).

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