11. (adapted from 1.6 8) Prove the characteristic polynomial of matrix A - is p(x) =...
Suppose A is a symmetric 3 by 3 matrix with eigenvalues 0, 1, 2 (a) What properties 4. can be guaranteed for the corresponding unit eigenvectors u, v, w? In terms of u, v, w describe the nullspace, left nullspace, (b) row space, and column space of A (c) Find a vector x that satisfies Ax v +w. Is x unique? Under what conditions on b does Ax = b have a solution? (d) (e) If u, v, w are...
Problem 3. Suppose A has eigenvalues 0, 3, 5 with corresponding independent eigenvectors u, v,w. (a) Give a basis for the nullspace and a basis for the column space. (b) Find a particular solution to Ax=y+w. Also, find all solutions to Ax=y+w.
Q1. Let A = be a 2 x 2 matrix. 30 (a) Find the characteristic polynomial of the matrix A. (5 pts) (b) Find all eigenvalues and associated eigenvectors of the matrix A. (10 pts) (c) If is an eigenvalue of A, what do you think it would be the eigenvalue of the matrix 7A?(Justify your answer) (5 pts)
(1 point) Find the characteristic polynomial of the matrix 5 -5 A = 0 [ 5 -5 -2 5 0] 4. 0] p(x) = (1 point) Find the eigenvalues of the matrix [ 23 C = -9 1-9 -18 14 9 72 7 -36 : -31] The eigenvalues are (Enter your answers as a comma separated list. The list you enter should have repeated items if there are eigenvalues with multiplicity greater than one.) (1 point) Given that vi =...
Find the characteristic polynomial and the eigenvalues of the matrix. 8 7 -7 - 6 Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for 3 x 3 determinants. [Note: Finding the characteristic polynomial of a 3 x 3 matrix is not easy to do with just row operations, because the variable A is involved.] 500 -7 3 8 - 5 0 4
2. (-16 Points) DETAILS CHENEYLINALG2 6.1.017. 0/2 Submissions Used 9 Let A - Find the characteristic polynomial. 11 Det(A - AI) - Find the eigenvalues and eigenvectors for each eigenvalue. (Order your answers from smallest to largest eigenvalue.) 21 has eigenspace span 12 has eigenspace span Find a matrix P such that p-'AP is a diagonal matrix. P
3. You are given the following matrix -4 12 2 7 a)4 points) Find a basis for the nullspace of (b) 4 points] Using the columns of A, find a basis for the column space of A (c) [2 points What are the dimensions of these spaces? (d) [2 points] ls the vector u-I1-1 0 ојт in the nullspace of A? Why? (e) [4 points] Is the vector w-17-9 9-9]T İn the column space of A? If so, express w...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
How can I get the (a) 3*2 matrix A? x 7. [30pts] Let V be the subspace of R consisting of vectors satisfying x- y+z = 0 y (a) Find a 3x2 matrix A whose column space is V and the entries a a1 0 = (b) Find an orthonormal basis for V by applying the Gram-Schmidt procedure (c) Find the projection matrix P projecting onto the left nullspace (not the column space) of A (d) Find an SVD (A...
Linear Algebra Problem Complete each of the following. (a) Suppose that A has characteristic polynomial p(\) = 13(2+1)5(1-3)4. In a table, list the eigenvalues of A, along with their algebraic multiplicities. Using this, find the order of A. (b) If A has a ten-dimensional column space, what is the nullity of A? Is A diagonal- izable? Explain.