3.Given a nonstandard basis of a vector space of a second-order polynomial asB = { 1 + x, -x + x², 1 + x − x²}. If a transformation T: P₂ R³ is given asT(¹+x) - ().T-*+ **)-().ra+x-)-(3)=; T(−x x²) = 2;T(1 x²) =Determine T(1 - x + x²)

Answered: Image /qna-images/answer/d0a8e7d0-c027-4f74-8e8f-ad03d0b4a555.jpg

Answered: 3. Given a nonstandard basis of a…

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

Show that the transformation T defined by T(x₁, x₂) = (2x₁ - 3x₂, X₁ +5, 5x₂) is not linear. If T is a linear transformation, then T(0) = T(0,0) = (2(0)-3(0), (0)+5, 5(0)) -0.00 0 = 0 and all scalars c, d. (Type a column vector.) Check if T(0) follows the correct property to be linear. and T(cu + dv)=cT(u)+dT(v) for all vectors u, v in the domain of T Substitute. Simplify.
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.
Suppose T: M22→P2 is a linear transformation whose action on a basis for M2 2 is as follows: 2 -2 0 0 0 3 11 -4x2+12x-4 T| x2 4x 3x2 – 12x+3 T = 2x2 - 12x-1 0 -2 1 0 0 3 12 Describe the action of T on a general matrix, using x as the variable for the polynomial and a, b, c, and d as constants. Use the 'A' character to indicate an exponent, e.g. ax^2-bx+c. a = 0 lса
Let P3[x] be the vector space of all polynomial of order at most 3. Suppose that B = { 1, (x-1), (x-1)2, (x-1)3}. Show that B is a basis for P3[x]. Give p(x) = x3+ x, find [p(x)]B.
,
[3.3] Show that (C,+) and (R2, +) are isomorphic.
4. Let B = {v1, v2, V3} and C = {w1, w2, w3} be bases for R3, with vectors defined below. -(:) -() --() v1 = V2 = V3 = and --(:) --E) --() 1 wi = , w2 = 1 , W3 = 1 Let L: R³ → R³ be the linear transformation defined by L: Find [L]B and [L]c, the matrices associated to L with respect to B and with respect to C.
find a basis for the range from r2 to r3 as defined by [3x1+4x2; x1-2x2; 4x1]
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.
(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

3. Given a nonstandard basis of a vector space of a second-order polynomial as R³ is given as B = {1 + x, -x + x²,1 + x - x²}. If a transformation T: P₂ T(1 + x) = >-();r-x+x) - ()+*-*- (1) (19) |; x²) = x²) = ; T(1 + x - x²) = Determine T(1-x+x²)

3. Given a nonstandard basis of a vector space of a second-order polynomial as R³ is given as B = {1 + x, -x + x²,1 + x - x²}. If a transformation T: P₂ T(1 + x) = >-();r-x+x) - ()+-- (1) (19) |; x²) = x²) = ; T(1 + x - x²) = Determine T(1-x+x²)