Let T be a linear transformation from R to P2 defined by-(E)T=(-2a1 + a2 + 4a3)x² + (a1 – 7a2)x + 8a2 – 5aza3Let Bbe a basis for R3 and B' = {x², x, 1}be a basis for P2a11a12a13Find the standard matrix A of T with respect to B and B' . If A =а21а22a23thenаз1а32a33a11 =, а12a13a21 =a22 =, a23аз1a32 =a33 =

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Answered: Let T be a linear transformation from…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If the matrix of change of basis form the basis B to the basis B' is A = (³) O A. OB. 04 (3) 00 (12) 52 21 then the first column of the matrix of change of basis from B to B is:
Let T and T2 two linear transformations from R? to R², defined by 7(;) - ) ()-() 3x - 2y (;) (} x+ y T1 T2 y - 3x for all (x, y)" e R?. Find the standard matrix representations for each of the following composite functions, and their inverse transformation, if they exist, by using the algorithm for matrix inversion in Lemma 3, Topic 14. )T followed by T2. (i) T2 followed by an anticlockwise rotation by angle 5 in R?. (ii) A reflection in the line y = 2x followed by a projection onto the line y = -3x in R2.
Let T be a linear operator defined on P3 by T(a0+a1x+a2x^2)=a0+a1(x-1)+a2(x-1)^2. The third row of the matrix associated with T with respect to base 1, x, x^2 is: a) (0,0,1) b) other response c) (1,-2,1) d) (1,0,0) e) (-1,1,0)
a) Verify the given sequence B is a basis ; and b) compute the images of the standard basis vectors with respect to the given linear transformation. 15. fi = 1+2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1 + 4x – x², B = (f1, f2, f3). - 1 T(fi) = (; 10)- 4 T(fi) = (; ).T ,T(f2) : 3 8
9. Consider the following 2 x 2 matrix A and basis S of R²: A ¹ = [² %) and 5 = {[_¹₂] · [_³;]} The matrix A defines a linear operator on R². Find a matrix B that represents the mapping A relative to the basis S.
7-8. Determine whether the space of the matrix A 2- I 2 O > Am 4 vector ū belongs A₁ = 0 to null
33. Prove that the row vectors of an n x n invertible matrix A form a basis for Rn.
1. Let u₁, U2, U3 } and { V₁, V2, V3 } be ordered bases for R³, where H and U₁ = 2 1 U₂ = 1 V1 = 1 -1 -2 U3 = -8-0-0 V2 = 1 V3 (a) Determine the transition matrix corresponding to a change of basis from the ordered basis {u₁, U2,, u3} to the ordered basis {V₁, V2, V3}. (b) Write vector z = 2u₁ - 3u₂ + u3 as Use this transition matrix to find the coordinates of with respect to {V₁, V2, V3}.
of Let W be the set of all 2 × 2 matrices [a b] C d basis for W is S = (a) • X (b) following questions: • Y = • r {[ || (c) where, c = 0 and d = 2a - 3b. A || •S= 1 x ] []} " S О У I dim(W) - Answer the
21. Suppose that A {a1,a2} is a basis of the vector space V = R? and B, = a1 + 2a2, B2 = 201 + 5a2. (a) Prove that B = {B,,B,} is also a basis of V. %3D (b) Find the following coordinate matrices : [B1]A, [B,]B, [a1]a and [a2]B. (c) Find the transition matrix (=Change of basis matrix) from A to B
The vectors -6 01 = 02 = 03 =| 4 form a basis for R' if and only if k +
Let L: R3 R2 be defined by the following. X1 + X3 ***] 8x2 Suppose X1 L X₂ P = = 1 -(HH) 4 3 3 1 2 B₁ = 4 ={[8][;}} is an ordered basis for the domain and B₂ = is an ordered basis for the range. Find the matrix representation P for L relative to B₁ and B₂ such that [L(u)]b₂ = P[u]ß₁'

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