→ R² byProblem. Define a linear transformation T: P2T(P) = [P].Find a polynomial q in P₂ such that Span{q} is the kernel of T (justify youranswer, of course), and prove that T is onto.

Answered: Image /qna-images/answer/2e0beb22-480d-402e-93ef-a89ba0da5d48.jpg

Answered: Problem. Define a linear transformation…

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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Problem. Consider the polynomials P1, P2, P3 EP₁ defined by P₁(t) = 1+t P₂ (t) = 1 t P3(t) = 2. By inspection, write down a linear dependence relation among the three polynomials. Then find a basis for W = Span({P₁, P2, P3}). Is it the case that W = P₁? Explain your answer.
3. Suppose T: M22 P3 is a linear transformation whose action on a basis for M22 is as follows: 3- = T T = -x³ + x²-3x+2 = X T T :-3x² - x-6 = -4x³8x² 10x - 16 Determine a matrix, Ar that takes the matrix entry vectors to the coefficient vectors of the polynomials. Then determine a basis for the kernel and image of T. Is T one-to-one and/or onto? If it is not one-to-one, show this by providing two matrices that have the same image under T. If T is not onto, show this by providing a polynomial in P3 that is not in the image of T and show that it is not in the image of T.
For each of the following linear transformations T, determine whether Tis invertible and justify your answer. (a) T: R2 R3 defined by T(a1, a2) = (a12a2, a2, 3a1 + 4a2). (c) T: R3 R3 defined by T(a1, a2, a3) = (3a12a3, a2, 3a1 + 4a2). (d) T: P3(R) P2(R) defined by T(p(x)) = p'(x).
Suppose that T is a linear transformation from R" to Rm where m
Find the Kernel of the following linear transformation. Also find the Ker(T). (i) T: P3 → P2, where T (ao + A₁x + A²x² + A3x³) = a₁ + 2a2x + 3a3x² (ii) T: R² → R², where T(x, y) = (x − y, y — x)
Find a basis for the kernel and the image of the linear transformation T: R³ R³ given by T(x) = Ax, where 1 2 3 A= -1 -2 -3 13 4
Suppose T: P2→R is a linear transformation whose action on a basis for P2 is as follows: 6 -8 T(x2+x+1) = -1 T(x²+x+2)=|-6 X 2 4 -4 | Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two polynomials that have the same image under T. If T is not onto, show this by providing a vector in R that is not in the image of T. Use the A' character to indicate an exponent and x as the variable for polynomials, e.g. 5x^2-2x+1 T is not one-to-one: T(0)= 0 and T(0)=|0 T is onto
If T : R4 [a] → M2x3 (R) is a linear transformation which of the following can not be the rank of T? 3
1 T:R? + R be defined by T X2 X1+X2 Then T is a linear transformation. [x1-X2] True False

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Problem. Define a linear transformation T: P2 T(P) = [P(3]. R* by Find a polynomial q in P₂ such that Span{q} is the kernel of T (justify your answer, of course), and prove that I is onto.