II Coordinates5. Fix some a € R, and define polynomialsPo(X), p1 (X), p2(X) = P(R) byPo(X) = 1, p₁(X)=X+a, _p2(X) = (X + a)²(a) Prove that B = {Po(X), P₁(X), p2(X)} is a basis of P₂ (R), thespace of all polynomials of degree < 2 with real coefficients.(b) For any f(X) = co + C₁X + c₂X² = P₂ (R), compute thecoordinates off with respect to the ordered basis B from part (a).

Answered: Image /qna-images/answer/955a0014-0843-485c-9b43-0e5ac9e5bd20.jpg

Answered: II Coordinates 5. Fix some a € R, and…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

Draw the projection of b onto a and also compute it from P = xa, then construct the projection matrices PI and P2 onto the lines through the a's
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.
2 Let T: R² R² be defined by T Let u = T(u) · [²] = {√₁ = [1] ₂₁ [2]} H ₁ and V2 = X1 ([2₂]) = [ X2 = X1 Ex: 5 Find T(u), the image of u under T. Find [T(u)]c, the coordinatization of T(u) with respect to the basis C. X1 [T(u)]c = - x2
Let S P₂ P₂ be the linear transformation where S(x) = −12x − 30 and S(1) = 5x + 13. 0 Suppose Mg(S) = [-2] MB 0 3 Let Ɛ = {x, 1} be the standard basis of P2. Then P&B = 1 1 Then B = = {x+ 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.
Consider the basis B₁ = {(1,2), (3, -1)} of R2. Define T: R² = {(1, 0, 0), (1, 1, 0), (1, 1, 1)} of R³ and the basis B₂: R³ such that T((a, b)) = (a + 2b, - You may assume that T is linear. -a, -b). a. Find the matrix M(T) for T with respect to the standard bases in the domain and codomain. b. Find the matrix M(T) for T with respect to B₂ in the domain and B₁ in the codomain.
This is a question from a linear algebra course: Let V = R[X]3, the polynomials of degree at most three, and B = {1, X, X2, X3}. Show what the image under fB is of:• the four basic elements: P1(X) = 1, P2(X) = X, P3(X) = X2 and P4(X) = X3• P(X) = 2 + 6X + 3X2 + 4X3
find a basis for the range from r2 to r3 as defined by [3x1+4x2; x1-2x2; 4x1]
Consider the basis B = {({2, 1), (3, 1}} for V = R2 and let {x1, x2} be the dual basis of B. Then for an arbitrary pair (c, d) EV, the set {x1(c, d), x2(c,d)) is equal to: {(3d-c), (c-2d)} {(3d+ c), (c+ 2d)} {(2d-c), (c + 3d)} {(2d+c), (c + 3d)} {(3d+c), (2c - d)}
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.
= [-16 12]. -9 Find bases for N(A), the null space of A, and C(A), the column space of A. Let A = Basis for N(A) is Basis for C(A) is
(a) Consider the real linear map T : R' - R', expressed as follows: (1 2 -1 4 9 -1 X1 + 2x2 - x3 X1 X1 X1 T| x2 4x1 + 9x2 - x3 i.e. T x2 X2 2x1 + 7x2 + 7x3) X3 2 7 7 X3 (i) Find a basis for the kernel of T. (ii) Find a basis for the image of T. (iii) Does the image of T include every vector in R'? Justify your answer. (You are not required to verify that your answers to (a) (i), (a) (i) are bases.) (b) Consider the following subset of R: X1 S = x2 ER: x2 + 4x3 = 2 Does there exist a real linear map, from R to R' say, such that the set S is the image of the real linear map? Justify your answer.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated

Numerical Methods for Engineers

Advanced Math

ISBN:

9780073397924

Author:

Steven C. Chapra Dr., Raymond P. Canale

Publisher:

McGraw-Hill Education

Introductory Mathematics for Engineering Applicat…

Advanced Math

ISBN:

9781118141809

Author:

Nathan Klingbeil

Publisher:

WILEY

Mathematics For Machine Technology

Advanced Math

ISBN:

9781337798310

Author:

Peterson, John.

Publisher:

Cengage Learning,

Basic Technical Mathematics

Advanced Math

ISBN:

9780134437705

Author:

Washington

Publisher:

PEARSON

Topology

Advanced Math

ISBN:

9780134689517

Author:

Munkres, James R.

Publisher:

Pearson,

SEE MORE TEXTBOOKS