3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n x n matrix A.a)Prove ker T C kerT² C kerT³ C …..Note: ker T" = nullspace of A"b) If kerT = ker T2, show that in general we must have ker T" = ker Tn+l for all n> 1.c) Use this result ( i.e., ( b) ) to show that there is no 3 × 3 matrix X such that0 1 0]0 0 10 0 0x²

Let T be a linear transformation ℝn to ℝn given by Tv=Av, A∈Mn×nF To prove that…

3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n × n matrix A. а) Prove ker T C kerT² C kerT³ C ….. Note: ker T" = nullspace of A" b) If kerT = kerT², show that in general we must have ker T" = ker T"+1 for all n> 1. c) Use this result ( i.e., ( b) ) to show that there is no 3 x 3 matrix X such that Го 1 01 x2 = 0 0 1 0 0 0

3. Let T be a linear transformation of R" to R" given by T(v) = Av for some n × n matrix A. а) Prove ker T C kerT² C kerT³ C ….. Note: ker T" = nullspace of A" b) If kerT = kerT², show that in general we must have ker T" = ker T"+1 for all n> 1. c) Use this result ( i.e., ( b) ) to show that there is no 3 x 3 matrix X such that Го 1 01 x2 = 0 0 1 0 0 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

4. Let T-RR' be the lineor trons.form such that T (xiyiz)=(yt7, X+y, X+Z), By considering matrix representating T with re spect to basis you like, answer the following questionss any * T is inver tible linear transformatione Yor N? 3. For every b ER, the equation T(P)=b is solvable Y or N ? vector solution. * The equation TR)= has nonzero Y or N? for As ģ : 7. let S:RR be the by s? ) = A? Now 4 lincar trous formation defined the the composition So T-R R and auswer following pue. Consider S.T is c înver tible line ar trans formation, Y or NS an A for every BER the eguation S. T(?)=B is solvable the only Solution of the equation. S. TQ)= 8 vector is N? or
1. Let A = MB (f) be the matrix associated with the linear map f(x,y.z) = (3x-y+4z.y+z-x.x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,-5)} and B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0)) Then the first column of A is: OA-16 OB. 4 17 17 B 0 (3) 13 1 OC. 3 O D./ O E. 7-
17. Let T R² → R² be a linear transformation that maps 5 2 into and maps v = 2 3 3 Tis linear to find the images under T of 3u, 2v, and 3u + 2v. u= into . Use the fact that
could you pleaae do all parts and could the answer be written out please , not typed up. Thank you
7. Let T: R³ R³ be defined by (a) (b) (E) x1 + x2 + x3 = X1 X2 X3 [X1X2 X3] Find the matrix of linear transformation for T. T Find all vectors 7 € R³ such that T(x) = x.
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
7. Consider the set of vectors S = {u = (-1, 2, – 1), v = (2, – 5, 0), w = (1, – 3, – 1)}, then find the vector orthonormal projection of "u" on "v" and the vector orthonormal projection of "u" on "w" respectively.
2. Let a = 2 ---- X= -2 4 and y = 4 2 (a) Find P, the projection matrix along the span of a. (b) Find proja (x). (c) Find proja (y). 3. Let A be a nonsingular 2 x 2 matrix. (a) Find P, the projection matrix onto C(A). (b) Find C(A) and comment on the result of part (a).
6. Determine the matrix of the linear mapping defined by first mirroring the space vectors in the line with equation (x, y, z) = (2-2+, -2+2+, 1-t) and then rotating 풍 them by the angle II around the x-axis in the positive direction.