2. Consider the transformation T : R4 → R³ defined byT(x1, x2, x3, x4) = (4x1 + x2 − 2x3 - 3x4, 2x1 + x2+x3 - 4x4,6x1 - 9x3+9x4)(a) Find the matrix associated with this transformation.(b) Find a basis for ker(T).(c) Find a basis for Range (T).(d) Is T one-to-one? Why?(e) Is T onto? Why?

"Since you have posted a question with multiple subparts, we will solve the first three subpartsfor…

Answered: 2. Consider the transformation T : R¹ →…

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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Let T : P2 → M2x2 be the linear transformation defined by T(ax? + bx + c) |0 a (a) Find the matrix for T with respect to the standard bases of P2 and M2x2- (b) Find the matrix for T with respect t B basis for M2x2. {1, x+1, x²+2x+1} and the standard
Let T : R³ → R³ be linear transformation. The sequence O ) (v1, v2, v3) = ( |) is a basis of R3. If T(v1) = |2),T(v2) = | 3,T(v3) = determine the standard matrix of T.
1 Let T : R² → R² be the transformation that reflect points across y = x. B is a basis in R². Find the matrix of T relative to B. 2 Find the 3- matrix [T]å of the transformation T : x → Ax, where (=123)₁ (2) B = {b₁,b2}, A = b₁ = " b2 = (1¹). = {(1).(+)}
Find the standard matrix of the linear transformation T: R² → R³ given by T(x₁, x2) = (2x₁ - 3x2, x₁ x1 1 1 [23 -3 -2 0 1 0 0 2 1 0 -2 0 0 2 -3 1 -2 2 -3 H 1 -2 1 -3 Го 2x2, x₂).
Show that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.
Suppose S: R² → R² and T: R² → R² are linear transformations given by 3x y (0)-[-1-(C)-[22] T -x-3y] (a) Find the standard matrix for S (assuming the stand basis for R³ and for R²). (b) Find the standard matrix for T (assuming the standard basis for R² and for R³). (c) Show that S and T are invertible. (d) Show that T is the inverse of S. (e) What is the standard matrix for T(S(z)) = To S(1), where I = R²?
[2] such that 7(6₁)=56, +46, and 7(6₂)=35, +56₂. Let == B= and ba = (a) The 28-matrix of T (in other words, the matrix of T relative to the basis 33) is 5 3 A= The set = {1,5} is a basis for R². Let T: R2 R² be a linear transformation → 5 (b) The standard matrix of 7 (in other words, the matrix of T relative to the standard basis for R²) is -1 You have answered these types of questions (in b) before. Can you now use coordinates and the A-SBS-¹ framework? Which technique do you prefer?
3x1-x2 (E) 1. Let T : R? → R³ be the linear transformation defined as T 2x1 + x2 %3D -X1 - X2 (a) Find its matrix representation. (b) Find T(0). (c) Find T-(0). 1 (d) Find T-
Let T:V → V be the transformation whose matrix relative to basis {b1, b2, b3} is Гх 3 -5 [T]B = ly 7 and T(b1 – 3b3) = 11b1 – 2b2 then [T(b1)]B %3D 2 -3 1 (a) -2 (b) -3 (c) -2 (d) 2 1 a e. None of these O b
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'
For a linear transformation L : R² → R³ is L(2, 1) = (2, 6, 8) and L(0, 1) = (2,0, 4). a) What is L(1,0)? b) What is the matrix representation of L? c) Use the matrix representation to calculate L(3,-1).
7. (a) Consider the ordered basis ß = [1,1 +x, 1 + x + x²] for V = P₂ and F = R. Suppose is in P2 and you know these coordinates: [] = (1, 2, 3). Find the vector 7. (b) Let A be a 2 x 2 matrix and a be a real number. Prove that det (a A)=a² det(A)

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