10. Let ø: Q(/2) → Q(v2) be the map given by ø(a+b/2) = a-bv2.Show that o is a Q-linear field automorphism.By Eisenstein criterion, x² – 2 is irreducible and hence {1, y2) isa basis of Q(V2) over Q.

Given : ϕ : ℚ(2)→ ℚ(2) be a map defined by ; ϕ(a+b2) = a-b2. To prove : ϕ is a ℚ-linear field…

Answered: 10. Let ø: Q(V2) → Q(/2) be the map…

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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Let F be a field and let be the linear transformation defined by a 13 A. B. a: F³ → F[X] → (a + b)X + (a + c)X5. Find the kernel of a. Find the nullity and rank of a.
The approximated value for e using the Maclaurin series expansion considering a true relative error that is conforming to one significant figure is: O a. 4.052 O b. 3,885 Oc3414 O d. 2414
3 be W be vector a positive integer, V and field F, T: VV be linear transformation, and Spaces over f: V²->w be a multilinear alternating function. Let g: V" w be the function defined by Let n a V g (V., ..., Vn ) = + (T(V₁), T(V₂), ...,T (Un)) (V.... . Vn) EV" Prove the following statements. (a g is multilinear and alternating. (b) If V=F", W=F₁, and T=LA for some A € Mnxn (F), then g= (detA/.f
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Please help with both parts b and c: (will leave a like)
+b] Let T : R? → R be a linear transformation. Which of the following is equal to T a +b O A. T (B) +" (!) В. T +T +T +T Oc. T 피 (B) OD. | 2T
Show that linear mapping T : P2 (R) → R3defined by T (ax2 + bx + c) = (a + b, a + c, b+ c), is an isomorphism.
Let ß and y be the standard ordered bases for R" and R", respectively. For the following transformations T: R* → R", compute [T]}. (а) Т: R? — R3 defined by T(а,, а2) — (2а, — а,, За, + 4а,, а). (b) T: R3 (c) T: R3 (d) T: R3 T(а, а,, аз) — (2а, + aз, — а, + 4a, t 5аз, а, + a). (е) Т: R" > R" defined by T (ај, а2, .., а,) 3D (ај, ај, ...,a). (() T:R" — R" defined by T(aj, аz, ..., an) 3 (ал, а, - 13 .., аj). (g) T: R" - R? defined by T(а,, а, аз) — (2а, + За, — аза а, + ag). > R defined bу T(а;, а2, а,) — 2а, + a, — Заз- R3 defined by R defined by T(a1, a2,..., a,) = a1 + a„.
4. Let VCR" be a subspace. Let F: V→V and G: V → V be invertible linear transformations. Denote by F¹: V→V and G¹: V → V the inverses of F and G respectively. Is the map H: V → V defined by H(v):= F(G(F¹(G¹(v)))), for v € V, a linear transformation? Justify your answer.
5. 3) Let : R₂[x] → R³ be a linear transformation such that ( (1) = = l(x): = 0 and l(x²) = = 1 (a) Find a formula for l(a + bx + cx²). (b) Show that is a vector space isomorphism and find a formula for l-¹.
Show that T is a linear transformation

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10. Let ø: Q(V2) → Q(/2) be the map given by ø(a+bv2) = a-bv/2. Show that is a Q-linear field automorphism.