ⒸFind an orthonormal basis [b1,b₂, b3] of spam [3]Lohere B={(1,0,1), (1, 0,-1), (0,1,2)}, using theGram-Schmidt orthonormalization process.Make sure that b₁ = 1 (1,0,1).

Answered: Find an orthonormal basis [b₁,b2, b3]…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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3. If A = 0 -1 0then the bases for the eigenspaces of A2 + 21 are: 2. a) ({, = 7, v, = (1,0,0), {A2 = 4, v (-1,0,1)}, and (A = 1, v, = (-1,6,0}) b) {{ =-3, v, = (-1,6,0}), (A = 3, v (1,0,0}}, and (A = 0, v, = (1,0,0}] c) [{ = 27, v, = (1,0,0},{22 = 6, v = (-1,0,1}}, and (As = 3, s =(-1,6,0}} d) {{2 = 2, v, ={-1,0,1}}. [A2 = 23, , = [1,0,0), and (As =-1,, {-1,6,0}}
Consider B = {1+x,x + x²,1 + x²}, B' = {1+x²,1 – x²,1 – 2x}. i) Show that B and B' are bases for P2. ii) Find the coordinate matrices of p(x) = -x² + 4x + 7 relative to B and B' %3D iii) Find the transition matrix PB--B'- iv) Verify that [x(p)]g' = Pg¬g'[x(p)]g-
The eigenvalues of a 4x4 matrix, A, are 5, 3, 3, and 1. Execute Gram-Schmidt process on the following two vectors to find an orthonormal basis set: Vị (2,2), v2 = (5, 0). After normalization, we have two new (x1, Y1) and (x2, Y2). vectors u1 U2 What is the trace, i.e. sum of the diagonal entries, of the matrix A? What is the second largest eigenvalue of A??
3.) Apply the Gram-Schmidt orthonormalization process to the basis B = {1,t, t²} in P2 using the inner product 1 (p, q) = | p(t)q(t)dt -1 Show all calculations.
Find the characteristic polynomial, the eigenvalues and a basis of eigenvectors associated to each eigenvalue for the matrix -3 0 ^= (-2) A 9). a) The characteristic polynomial is p(r) = det(A - rI) = (r+3)(r-4) b) List all the eigenvalues of A separated by semicolons. -3;4 c) For each of the eigenvalues that you have found in (b) (in increasing order) give a basis of eigenvectors. If there is more than one vector in the basis for an eigenvalue, write them side by side in a matrix. If there is only one eigenvalue, enter the zero vector as an answer for the second eigenvalue. i) Give a basis of eigenvectors associated to the smallest eigenvalue. ab sin (a) ∞ a ii) If there is another eigenvalue, give a basis of eigenvectors associated to this eigenvalue. Otherwise, write the null vector. 08 f C X
If T is a normal operator over C then it has a spectral decomposition + AkTk. Show that if T* ci for a real number c). T = A1T1 + -T then every eigenvalue is purely imaginary (i.e.
7. Let A be an n x n Hermitian matrix. (a) Show that (Au, v) = (u, Av) for any n-dim vectors u and v. Here (, ·) denotes the inner product. (Hint: rewrite the inner product as matrix multiplication.) (b) Let x be an eigenvector associated to the eigenvalue A. Based on part (a), show that A(x, x) = X(x, x). (c) As in part (b), show that = X; that is, the eigenvalue A is real. 1

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Find an orthonormal basis [b₁,b2, b3] of spam[13] where B = {(1,0,1), (1,0,-1), (0₂1. 2)}, using the Gram-Schmidt orthonormalization process. Make sure that b₁ = 1 (1,0,1).