2. For a linear transformation T: RR³ defined byT(w,x,y,z)=(w+y-z,x-y+z,w+x), which of the following vectors belong tothe kernel of T? Circle all appropriate responses and show all your work!a. (1, 1, 1, 1)b. (0, 0, 1, 1)c. (1, 0, 0, -1)d. (0, 0, 0, 0)

Answered: Image /qna-images/answer/2533c13b-58c5-4188-947b-9932a0b95b2c.jpg

Answered: 2. For a linear transformation T: R →…

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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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.
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
Determine if the linear transformation T: R R3 defined as T(r, y, z) = (x+y+ 3z, 3a+ 4y + 7z, +y+2z) is invertible or not. For another transformation defined by the formula T'(x, y, z) = (x +y+ z, y-3z, z) write the formula for both ToT' and T' oT.
Let D be a parallelogram with vertices (0,0), (2, 1), (2,3), (4,4) Find a linear transformation x=x(u,v), y=y(u,v) and use to evaluate (x + y)dA 35 2 a. b. C. d. e. 16 32 3 12 15
What is S'?
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) =
Suppose T: R¹ → R™ is a linear transformation and {V₁, V2, V3}) is a set of vectors in R". (a) Complete the following definition: The set {V1, V2, V3} is called linearly independent if (b) (c) T is one-to-one if and only if Ker (T) = Show that if T is one-to-one and {V₁, V2, V3} is linearly independent, then the set {T(V₁), T(V₂), T(V3)} is linearly independent.
Suppose T is a linear transformation, where T(u) T(7) (3, – 5) (1, 0, 0) (0, 1, 0) (0, 0, 1) (5, 4) 6- T(w) (5, 4) Find the following: T(5u + 70 + 2w) (Enter vectors (x, y) using ) T(( – 3, 5, 4)) = (Enter vectors (x, y) using ) -

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2. For a linear transformation T: R → R' defined by T(w,x,y,z)=(w+y-z,x-y+z,w+x), which of the following vectors belong to the kernel of T? Circle all appropriate responses and show all your work! a. (1, 1, 1, 1) b. (0, 0, 1, 1) c. (1, 0, 0, -1) d. (0, 0, 0, 0)