(a) (2 points) Describe the eigenvectors and eigenvalues of the linear transformation : C'(R) + C°(R)....
s={(8.60) :) :) is a basis of M3x2(R)? (d) (1 point) The set = {(1 9:(. :) : 6 1) (1 1) (1 :) :()} is linearly independent. (e) (1 point) For a linear transformation A:R" + Rd the dimension of the nullspace is larger than d. (f) (1 points) Let AC M4x4 be a diagonal matrix. A is similar to a matrix A which has eigenvalues 1,2,3 with algebraic multiplicities 1,2, 1 and geometric multiplicities 1,1, 1 respectively. 8....
please answer both a and b
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2-R2 be defined by f(x,y) = (y,z), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f. Hence, or otherwise, show that: a vector subspace U-o or...
Problem 2 (Eigenvalues and Eigenvectors). (a) If R2 4 R2 be defined by f(x,y) (y,x), then find all the eigenvalues and eigenvectors of f Hint: Use the matrix representation. (b) Let U be a vector subspace (U o, V) of a finite dimensional vector space V. Show that there exists a linear transformation V V such that U is not an invariant subspace of f Hence, or otherwise, show that: a vector subspace U-0 or U = V, if and...
2. (a) Let T be the linear transformation which projects R^3 orthogonally onto the plane 2x+3y+4z = 0. What are the eigenvalues and associated eigenspaces of T? Justify your answer. (b) Does the linear transformation described in (a) have an inverse? Why, or why not?
Problem 2. (18 points) (a) Find a fractional linear transformation that maps the right half-plane to the unit disk such that the origin is mapped to -1. (b) A fixed point of a transformation T is one where T(2) = 2. Let T be a fractional linear transformation. Assume T is not the identity map. Show T has a most two fixed points. (c) Let S be a circle and 21 a point not on the circle. Show that there...
1. Let T(2) := tbe a fractional linear transformation, as above (so a, b, c, d e C with ad-bc7 0). Argue that T is differentiable everywhere except one point and find T'(2). (Don't prove it directly - use results from class.) When is T' (2) = 0? What happens if the condition on coefficients is not met, i.e. if ad-bc=0? What can you say about the function in this case?
(d) (4 points) Let T : R² + Rº be the transformation that rotates any vector 90 degrees counterclockwise. Let A be the standard matrix for T. Is A diagonalizable over R? What about over C? (e) (3 points) Let T : R4 → R4 be given by T(x) = Ax, A = 3 -1 7 12 0 0 0 4 0 0 5 4 0 4 2 1 Is E Im(T)? 3 (f) (9 points) Let U be a...
4. (22 points) Let To : R2 R2 be the linear transformation that rotates each point in IR2 about the origin through an angle of θ (with counterclockwise corresponding to a positive angle), and let T,p : R2 → R2 be defined similarly for the angle φ. (a) (8 points) Find the standard matrices for the linear transformations To and To. That is, let A be the matrix associated with Tip, and let B be the matrix associated with To....
This is the question:
42 CHAPTER 2. BASICS Example 2.15 We consider the one-dimensional Sturm-Liouville eigenvalue problem (2.24) - u"(x) = \u()0<<<, (0) = u(T) = 0, that models the vibration of a homogeneous string of length that is fired at both ends. The eigenvalues and eigenvectors or eigenfunctions of (2.24) are x = k?, ux() = sin ka, KEN Let u" denote the approximation of an (eigen)function u at the grid point Ii, uiuti), Di=ih, 0<i<n +1, h =...
2. (10 points) According to the life-cycle hypothesis individuals care about the PDV of their lifetime income in making current consumption decisions. As a result in our short run model, we have assumed consumption to be characterized as C aY However, now suppose that we also more accurately incorporated the role of interest rates and rewrote the consumption equation as (a) Derive the new IS equation and show how the shape would be different from the original equation. (b) If...