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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Exercise 1. A student is asked to prove If m is even and n is odd, then m² +n² is odd. The student provides the following proof: Proof. If m is even and n is odd, then m² + n² is odd. Since m is even, m = = 2k for any integer k. Since n is odd, n = 2k + 1. Note that m² + n² = (2k)² + (2k + 1)² = 8k² + 4k +1 So m² + n² is odd because it is an even number plus 1. (a) Identify the errors in the proof above. There are multiple errors. Explain each error. Your explanation should be written to the student who made the error and should try to help the student understand why what they wrote is incorrect. (b) In addition to errors, provide suggestions to improve the student's proof writing. That is, explain what parts of the proof are unclear or unjustified. Explain how such parts of the proof prevent some students from fully following the ment made. (c) Finally, provide a correct proof of the theorem above.
2(3)
Lett: V→ V' be a linear transformation. Then (1) Ker (t) is a vector sub space of V, and (ii) im (1) is a vector sub space of V
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.
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.

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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)