7. Decide whether or not the following situations are possible and justify your answers. (a) (1...
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....
(1 point) The linear transformation T: R4 R4 below is diagonalizable. T(x,y,z,w) = (x – - (2x + y), -z, 2 – 3w Compute the following. (Click to open and close sections below). (A) Characteristic Polynomial Compute the characteristic polynomial (as a function of t). A(t) = (B) Roots and Multiplicities Find the roots of A(t) and their algebraic multiplicities. Root Multiplicity t= t= t= t= (Leave any unneeded answer spaces blank.) (C) Eigenvalues and Eigenspaces Find the eigenvalues and...
5. Consider the matrix A-1-6-7-3 Hint: The characteristic polynomial of A is p(λ ) =-(-2)0+ 1)2. (a) Find the eigenvalues of A and bases for the corresponding eigenspaces. (b) Determine the geometric and algebraic multiplicities of each eigenvalue and whether A is diagonalizable or not. If it is, give a diagonal matrix D and an invertible matrix S such that A-SDS-1. If it's not, say why not.
1. Determine whether the followings statements are true or false. (Com- ment: no reason needed.) (a) If the vectors ū1, ū2, üz are linearly independent, then the vectors ū1, ū2 are linearly independent as well. (b) The set {1,1 + x, (1 + x)} is a basis for P2. (c) For every linear transformation T: RM + R", there is an m xn matrix such that Tū = A✓ for all ū in R”. (d) If w1, W2 are vectors...
1. Determine whether the following set is linearly independent or not. Prove your clas a. [1+1, 2+2-2,1 +32"} b. {2+1, 3x +3',-6 +2"} 8. Let T be a linear transformation from a vector space V to W over R. . Let .. . be linearly independent vectors of V. Prove that if T is one to one, prove that (un)....(...) are linearly independent. (m) is ) be a spanning set of V. Prove that it is onto, then Tu... h...
For each of the following functions, determine whether or not they are (i) one-to-one and i) onto. Justify your answers (a) f : R-{0} → R and f(x) = 3r-1/x (b) g : R _ {1} → R and g(x) = x + 1/(x-1) (c) l : S → Znon-reg and l(s) = number of 1's in s, for all strings s E S, where s is the set of all strings of O's and 1's. (d) 1 : S...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
1. Decide whether each of the following is an inner product space. Justify your answers. (i) V = Mnxn(R) with (A, B) = tr(AB). (ii) V = M2x2(C) with (A, B) = tr (iii) V = P(R) with (f,g) = f(1)g(1). (iv) V = P(R) with :((1 ;-) B-4). (v) V is the collection of continuous functions from (0, 1) to C, and (5.9) = 'rg() dt. 4.s)-(sat).
(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...
1) Determine the critical points of the following function and characterize each as minimum, maximum or saddle point. See the attached slide. f(x1,x2) = x 2 - 4*x1 * x2 + x22 a critical point -, where f(x) = 0, if Hy( ) is Positive definite, then r* is a minimurn off. Negative definite, then r* is a maximum of . - Indefinite, then 2 is a saddle point of f. Singular, then various pathological situations can occur. Example 6.5...