Let F be a field and let V, be the F-vector space of polynomials in F[x] of degree

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Answered: Let F be a field and let V, be the…

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 Pn denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D : P3 - P2 be the function that sends a polynomial to its derivative. That is, D(p(x)) = p' (x) for all polynomials p(x) = P3. Is D a linear transformation? Let p(x) = a³x³ + ²x² + a₁ + a₁ and q(x) = b3x³ + b₂x² + b₁x + b₁ be any two polynomials in P3 and c E R. a. D(p(x) + q(x)) = . (Enter a3 as a3, etc.) D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = P3? choose b. D(cp(x)) = + c(D(p(x))) = = Does D(cp(x)) = c(D(p(x))) for all CER and all p(x) = P3? choose ✪ c. Is D a linear transformation? choose ◆
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
Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)
1. Let f: FF be a linear transformation with minimal polynomial m(x) = x²(x - 1)¹. (a) If dim(Vo) = dim(V₁) = 2, what are the possibilities for JNF (f)? (b) If the characteristic polynomial of f is c(x) = x¹(x-1), what are the possibilities for JNF (f)?
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.
Letf : R² → R be defined by f((x, y)) = −7x − 9y. Is ƒ a linear transformation? a. f((x₁, y₁) + (x2, Y₂)) = as x1, etc.) f((x₁, y₁)) + f((x2, 3/2)) = Does f((x₁, y₁) + (x2, Y2 )) = f((x₁, y₁ )) + ƒ((x2, y2)) for all (x₁, y₁), (x2, y₂) € R²? choose + b. f(c(x, y)) = c(f((x, y))) = = Does f(c(x, y)) = c(f((x, y))) for all c E R and all (x, y) = R²? choose c. Isf a linear transformation? choose . (Enter X₁
Let L be a linear transformation from P₂ → R2x2 such that L(f) — ƒ(1) 0 Find dim[Range(L)] ƒ(2) [ƒ(2) ƒ(0)
10. Let L: R R' be a map defined as follows: (4x1 +4 x2 - 2 x3 -9 x4 5x1 +2 x2 - 3 x3-9 x4 6x) – 4 x3 - 9 x4 7x1-2x2-5x3-9 x4) X2 X3 X4, (a) Show that L is a linear transformation on R'. (b) Write out the matrix S which represents L with respect to the standard basis. (c) Find a basis for Range S and Ker S. (d) State the "rank-nullity theorem" and verify explicitly that the result obtained in part (c) matches the statement of the theorem.
Consider the utility function u'(x1, X2, X3) = x²1 x³2 X3 . Find a monotone transformation of u'(x1, X2, X3) such that the new utility function is of the form u2(x1, X2, X3) = (X1º, X2º, x3º) Where a + b + c= 1.

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