→ a E Z.10. Let ø: Q(/2) → Q(2) be the map given by ø(a+bv2) = a-b2.Show that o is a Q-linear field automorphism.By Eisenstein criterion, x² –2 is irreducible and hence {1, v2} isa basis of Q(/2) over Q.e to search

We have given a map , ϕ : ℚ2 → ℚ2 such that , ϕa + b2 = a - b2 We need to show that , ϕ is an…

Answered: 10. Let o: 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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3) Let V = R[t]<2, W = M2x2 (R) with standard basis B = (1, t, t²), C = (m₁, m2, m3, m4). Suppose : V → W is some linear map such that, under the complexified basis Bc, Cc we have (a) (b) (c) [yc]c = 1 2 0 0 1 -1 22 2 1 4 0 What is yc(10 t²) What is p(1+t) (not the complexified function, the original) What is [y]
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„.
10. Let ø: Q(v2) → Q(v2) be the map given by ø(a+bv2) =a-bv/2. Show that o is a Q-linear field automorphism. By Eisenstein criterion, r² – 2 is irreducible and hence {1, v2} is a basis of Q(/2) over Q.
4. Define the following complex vector space = a € = (²x² = a +d=0} with the usual addition and scalar multiplication. (a) Find an appropriate value of r such that V~ Cr. (b) Give an explicit isomorphism T: V→ Cr. Make sure to prove that T is an isomorphism!
5. Let V be a vector space (note that this is an arbitrary vector space - not necessarily F"). Suppose dim(V) = n. Prove that a set of vectors (v₁,..., Vn} in V is linearly independent if and only if it spans V. • Hint 1: Remember (from sec. 1.6) that there is always an isomorphism A: V → F and isomorphisms "preserve" bases (and so preserves linear independence). Hint 2: Think in terms of pivots.
Find a norm of the operator A(x) = (3 + 2t2)x(t) in space C[0, 1]. а. 3 4 Ob. 7 2 С. 3 O d. 6 е. 4
Q4:A. Find vector field of { (1, 0), (1,1), (1, -1), -1, 0), -1, - 1)) For transformation f(x,y) = (1,2) اه العادق ندسة اللجنة
12. Let B = {1+2x, x - x²,x + x²}. (a) Show that B is a basis for P2. (b) Let p(x) = 1+ 3x + x². Compute the coordinate [p(x)]g of p(x).
14. Consider the basis B = {1 +x + x², x + x², x²}. for P2. Let p(x) = 2 + 3x + 2x². Then [p(x)]B 2 2 O None 1 2
7. Let V and W be finite-dimensional vector spaces and T: VW is an isomorphism. Prove that if 3 = {₁,.., Un} is a basis for V, then y = (T(₁) T(vn)} is a basis for W. 8. Let T R² R3 be a linear transformation such th that T(1,2) T(-1,3)= (1,4,2). Find T(x, y) and 7-1(x, y) if possible. = (0,3,-1) and
39. Find an orthonormal basis ū1, 2, 3 of R³ such that (ED) 2 and span (1) = span span (1, 2) = span 2 3
We define a inner product in = U₁V₁ + 2U₂V₂ + U3V3 in R³ for u = (U₁, U₂, U3) and v= (V₁, V₂, V3) members of R³. We also define S basis R³. S= = {W₁, W₂, W3} where w₁ = : (1,1,1), W₂ = (1,1,0), W3 = (1,0,0) a) Determine and explain whether the basis S is an orthonormal basis with the given inner product? b) If S is not an orthonormal basis, then determine the orthonormal basis S' (for the final step of normalization, if the number in the vector is difficult to simplify then just leave it in its initial form).

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