2. (a) Show that the list {p o(x),p 1(x), p 2(x),... ,P n(x),...,} wherePo(x) = 1, p1(x) = 1+x,P;(x) = 1+x ++ x³,•..)is a basis of the vector space F[x] of all polynomials with coeficients in F.(b) What are the coordinates of the vector x" relative to the basis (po(x),P1(x),..., P; (x),...)?

Answered: Image /qna-images/answer/ba29158c-e6f7-4b51-8890-e2401fc4f2ca.jpg

Answered: (a) Show that the list {p o(x), p 1(x),…

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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21. (a) Prove that for every positive integer n, one can find n +1 linearly independent vectors in F(-∞, ∞o). [Hint: Look for polynomials.] (b) Use the result in part (a) to prove that F(-∞, ∞) is infinite- dimensional. (c) Prove that C(-∞, ∞), C (-∞o, co), and C (-∞, ∞) are infinite-dimensional.
Find a basis {p(x), q(x)} for the vector space {ƒ(x) = P₂(x) | ƒ'(−3) = ƒ(1)} where P₂(x) is the vector space of polynomials in x with degree less than or equal to 2. p(x) = ₁ q(x) =
Let S = {1,2,3,4} and U = { ƒ € Fun(S) | ƒ(2) − 3ƒ(4) = 0, ƒ(3) = 0} Find a basis of U. Basis (U) = {} Write your basis using characteristic functions, with x1 for X₁ and x2 for X₂ etc.
Find a basis {p(x), g(x)} for the vector space {f(x) = P₂(1) | ƒ'(−2) = f(1)} where P₂(x) is the vector space of polynomials in with degree less than or equal to 2. |p(x) = , q(x) =
(b) Is the following set of vectors S basis for vector space V? Where (i) S = {(1,2), (0, 3), (1, 5)} and V = R? (ii) S = {(-1,3, 2), (6, 1, 1)} and V = R³ If not explain the reason.
Day traders typically buy and sell stocks (or other investment instruments) during the trading day and sell all investments by the end of the day. The following table shows the closing prices on September 22, 2015, of 12 stocks selected by your broker, Prudence Swift, as well as the change that day. Tech Stocks Close Change AAPL (Apple) $113.40 -1.81 ADBE (Adobe Systems) $84.66 1.34 EBAY (eBay) $25.61 -0.31 MSFT (Microsoft) $3.90 -0.21 S (Sprint) $4.40 0.02 WIFI (Boingo Wireless) $8.51 0.56 Non-Tech Stocks ANF (Abercrombie & Fitch) $21.81 -0.02 в (Воeing) $133.99 -2.03 F (Ford Motor Co.) $13.91 -0.40 GE (General Electric) $25.10 0.01 GIS (General Mills) $57.12 0.33 JNJ (Johnson & Johnson) $93.26 0.13 On the morning of September 22, 2015, Swift advised you to purchase a collection of three tech stocks and two non-tech stocks, all chosen at random from those listed in the table. You were to sell all the stocks at the end of the trading day. (a) How many possible collections are possible?…
In the vector space P (polynomial space) consider the linear operators D, A : P+ P defined by D(p(x))=p'(x) A(p(x))=xp(x) Find the sets Ker(D) and Im(D), Ker(A) and Im(A)
find a basis for the range from r2 to r3 as defined by [3x1+4x2; x1-2x2; 4x1]
Consider the operator Son the vector space by S(a+bx) = - a+b+ ( a + 2b) x A) Pick a basis B = { b₁,b₂3. Find the minimal polynomials NT, b, (X), Nr, 0₂ (x), and Ns (x) R₁ [x] given B) Show that S is cyclic by finding a vector v such that
Let P2(R) be the vector space of polynomials over R up to degree 2. ConsiderB ={1 + x − 2x^2 , −1 + x + x^2, 1 − x + x^2},B' ={1 − 3x + x^2, 1 − 3x − 2x^2, 1 − 2x + 3x^2}.(a) Show that B and B' are bases of P2(R).(b) Find the coordinate matrices of p(x) = 9x^2 + 4x − 2 relative to the bases B and B'.(c) Find the transition matrix P_B=→B'(d) Verify that, [x(p)]B'= P_B→B' [x(p)B].
3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.
1. Vectors v1 = (1,1,1,1,0), v2 = (-1,1,0,0,0), v3 = (1,1,-1,-1,0), v4 = (0,0,2,-2,0) and v5 = (0,0,0,0,2) form an orthogonal basis for a of R5 and X = (-3, -2,5,1,-4). Find the coordinates of X with respect to the given basis.

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