Let T be a linear transformation from R" to R$ .Determine whether or not T is one-to-one in each of the following situations:1. r > s2. r < s3. r = sA. T is not a one-to-one transformationB. T is a one-to-one transformationC. There is not enough information to tell

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Answered: Let T be a linear transformation from…

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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Consider the following three transformations from R³ to R³: x1 T x2 x1 - 2x3 2x1 - 2x2 - 5x3 x1 x1+5 x1 S x2 x2 R x2 = 0 X3 -x1 + 2x2+8x3) x3 x1x3 X3 Which of these transformations is/are linear? Select all correct choices. T S R None of these
5. A) Determine whether the given transformation is linear. STEP. T: R³ → R2x2 defined by (6) T = a1 2a2 + a1 a3 4a3a2
2. trace. Let n = 4 (we choose 4 for calculation simplicity, but the following are true for arbitrary n). Consider the following linear transformation: tr : R" +R a11 a12 a13 ain ... a22 a23 a2n ... a21 a32 a33 Ha11 + a22 + a33 + ...+ ann = ... a31 i=1 An3 ann ... anl an2 Let In be the n x n identity matrix (or equivalently, a vector in R""). Calculate (A) that tr(In) = n. Let A, B be two n x n matrices (or equivalently, a vector in Rn*). Calculate that (B) tr(AB) = tr(BA). Hint. Write A = [aj] and B = [br1]. Write down the diagonal of AB and BA in terms of aij and bri. Then compute tr(AB) and tr(BA).
38. a. Consider a linear transformation T from R³ to R. What are the possible values of dim(ker T)? Explain. b. Consider a linear transformation T from R4 to R'. What are the possible values of dim(imT)? Explain.
1. Let T be a linear transformation from M₂,2 A+ A+ A¹. M₂2. defined by the map Show that T is a linear transformation. What is its kernel?
For each question below, T: R" → R" where n = 2 or 3 such that T(x) = y. Which of the following transformations are linear? UA. B. C. OD. E. OF. Y1 Y2 Y1 Y2 Y1 Y2 Y1 Y2 Y3 Y1 = = = = Y2 Y1 Y2 = = 0 5x2 7x1 9 2x1 + x2 -X1 -3x1 4x1 6x1 x1 +8 X2 0 X1X2
Is T linear ? Is T one-to-one? Is T onto?

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Let T be a linear transformation from R" to R$ . Determine whether or not T is one-to-one in each of the following situations: 1. r > s 2. r < s 3. r = s A. T is not a one-to-one transformation B. T is a one-to-one transformation C. There is not enough information to tell