1. Recall that the image of the elementary vectors e; determines column i. Find the matrix thatrepresents the given transformation.(a) T: R² R² where e₁ →(b) T: R² R³ where ₁ →[B]2(c) T: R² R² where T→(d) T: R² R³ where TT(→123T(₂22₂)) = [I22112and e₂ →[³]4and e₂ →|35x1 + 3x₂2x1 + 4x2x1 + 3x₂2x1 + 4x23x1 + 5x2

Answered: Image /qna-images/answer/697f03d7-851b-4a3f-85a2-218896e2ad07.jpg

Answered: 1. Recall that the image of the…

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

Please solve all three parts
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.
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁0 T₂. T₁: R² R², T₁(x, y) = (x - 3y, 3x + 3y) T₂: R² → R², T₂(x, y) = (y, 0) A = A' = ↓ ↑ 00
Suppose T: R2-→R³ is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(-3U-2V). 14 18 U = V = T(U) = -7 T(V) =-9 3 16 21 T(-3U–2V) = | 0
Find a preimage of the vector 4 V= under the linear transformation T given by the matrix -3] = [44] -4 14 That is find the vector u such that T(u) = v. A U₁ ū₂
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
Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…
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
Suppose T: R³→R² is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(−U+2V). 2 U = 1 3 4 V = 3 4 0 T(-U+2V) = 0 0 -6 [:] 6 T(U)= -9 [3] 9 T(V) =
1. Let W be the line y = x in R². (a) Find a unit vector u on W. (b) Consider the linear transformation T: R² R² that projects each vector orthogonally onto W. Calculate the matrix for T which is uu¹. (c) Use that matrix to calculate T (d) Draw the vector [] [4] T[A] the line W, and the projection T

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS