Problem 5. Consider a linear transformation L : R³ → R² given byL(x, y, z) = (yx, z - y)for all (x, y, z) = R³. Find a matrix A such that L(v) = Av for every v € R³, where v andL(v) are regarded as column vectors.

We consider standard basis for R3 and R2. And find matrix required matrix A such that, L(v) = Av

Answered: Problem 5. Consider a linear…

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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2. The context of this question is linear transformations on the plane R². Let 7: R² R² be the function defined by first rotating the vector counter-clockwise by an angle of 45° and then stretching by a factor of 5 in the direction of the x-axis, and by a factor of 2 in the direction of the y-axis. Explain why T is a linear transformation and find the matrix associated to T.
Consider the linear transformation T: R3 R defined by T(x, y, z) = (x – 3y + 2z, 3x + y + 5z, -2x + 6y – 4z). Find the standard matrix of T.
Find the canonical matrix of the linear transformations below:
I1. In 1970 the Department of Natural Resources released 1000 splake (a crossbreed of fish) into a lake. In 1977 the population of splake in the lake was estimated to be 3000 Using a Malthusian law for population growth, estimate the population of splake in the lake in 1980. What would the estimate be for the year 1991 using the Malthusian law
IfT: R² → R³ is a linear transformation such that then the standard matrix of Tis (13) - 2 3 T([4]) = and T 26 10
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-
1. Let T : R² → R³ be a linear transformation such that T(e1) = u1 and T(e2) = u2, where e1, e2 8 is the standard unit vectors of R² and u1 3 and u2 = 2. Then find T 1 -2 6.
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
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (2x − 2y+z, x − 4y+z, x-y-z) relative to the bases B={u=(5,16,1); v= (2, 1, –4); w=(7,17,2)} and B'={u'=(2,0,1); v' = (1,0,1); w'=(3,2,0)) Then the second column of A is: *) 0 O A. (0 O B. "B) °) SOC. O D. (6)
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 O B. 13 1 4 17 17 °C) O D. (2 213 O E. "G) ņ m

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Problem 5. Consider a linear transformation L: R° →R given by L(x, y, z) = (yx, z - y) or all (x, y, z) € R³. Find a matrix A such that L(v) = Av for every v € R³, where v and (v) are regarded as column vectors.