-24-12.5-122-208-16-1is a matrix of a linear transformation T : RK → R".a. Find k and n.k =n =b. Find the dimension of the kernel (or null space) and range (or image).dim(Ker(T)) =dim(Range(T)) =%3Dc. Is T onto? choosed. Is T one-to-one? choose

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Answered: -2 4 -1 2.5 -1 2 2 -20 8 -16 -1 is a…

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 S P₂ P₂ be the linear transformation where S(x) = −12x − 30 and S(1) = 5x + 13. 0 Suppose Mg(S) = [-2] MB 0 3 Let Ɛ = {x, 1} be the standard basis of P2. Then P&B = 1 1 Then B = = {x+ x+ }
Let f : R? → R² be the linear transformation defined by -4 f(#) 4 -4 Let {(-1, –2), (3, 5)}, {(1,2), (1, 1)}, B C be two different bases for R?. Find the matrix [A% for f relative to the basis B in the domain and C in the codomain.
3. A transformation T:R² →R first projects each vector x= on y-axis, and then rotates -> vectors by . a. Find the matrix of the transformation T . b. Find the inverse transformation of T. if exists. -1 c. Draw the transformation T for x = 3
5. Find the standard matrix of the given linear transformation: (a) P: R? - R? projects a vector onto the line y = -2x. For the vector u = find P(ü). (b) Rg: R3 - R rotates a vector clockwise about the z - axis through the angle 30°. For the 21 vector u = 2| find R9 (ũ). 6. For the following linear transformations find S • T and T • S. Indicate the domain and codomain of the new linear transformations. x+ y+z] x-y-zl 2x + 7. Find the inverse of the following linear transformation: [3x + 2y] [5x + 3yl %3D
Q. 3. A transformation T: R² →R² first reflects vectors through the y axis, and then rotates vectors by 3 a. Find the matrix of the transformation T. b. Find the inverse transformation of T , if exists. -1 c. Draw the transformation T for x =| 2
. Linear transformation T: Rm →R" is given by one of the matrices below. Find m and n. 1 (a) [ 1 4 ] [ (b) 4 -2 3 8 i] 0 11 (c) 2-2 3 -3
5. y Let A be the linear transformation R² R2 that first reflects vectors across the line then rotates vectors 90 degrees counterclockwise abour the origin. Find the matrix representation of A. las 90⁰ >(y) [*] - [] CDC)
Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|
Consider a linear transformation T from R? to R² for which 7 (C) - [- ma 7 (1) - [- (H)-- T and T Find the matrix A of T. A: %3D
7. Find the standard matrix for the linear transformation T. T:R²→R², T maps e¡ to e1 +4e2 but leaves e2 unchanged.
(ED). = Let T -x +9y+ 3z - 2x + 4y + 5z 6x 8y + 2z Determine if the given function is a linear transformation. Find a matrix representation A such that T(x) Ax, or indicate that the transformation is not linear. O A = = O Not a linear transformation

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-2 4 -1 2.5 -1 2 2 -20 8 -16 -1 is a matrix of a linear transformation T : RK → R". a. Find k and n. k = n = b. Find the dimension of the kernel (or null space) and range (or image). dim(Ker(T)) = dim(Range(T)) = %3D c. Is T onto? choose d. Is T one-to-one? choose