Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients, viewed as a vector space over Q, and with the usual rules for vector addition and multiplication by scalars. Let T :V → V be defined by T(ax² +bæ +c) = (a+b+c)x+2(a+b+c). You may assume that T is linear. (a) Find a basis for the null space of T. (b) Find a basis for the range of T. (c) Verify the dimension theorem (rank-nullity theorem) for T.

Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients, viewed as a vector space over Q, and with the usual rules for vector addition and multiplication by scalars. Let T :V → V be defined by T(ax² +bæ +c) = (a+b+c)x+2(a+b+c). You may assume that T is linear. (a) Find a basis for the null space of T. (b) Find a basis for the range of T. (c) Verify the dimension theorem (rank-nullity theorem) for T.

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

(x1, y₁) + (x2, y2) = (0, 0) c (x, y) = (cx, cy) : Determine if R² with these operations is a vector space. Check each axiom to determine if it holds or fails. Show all work. 4. Consider R² with the following operations:
Let V be the vector space of functions spanned by B = {b₁ = 2 sin x + 9 cos x, b₂ = 4 sin x + 7 cos x} where x C = {C₁ = sin x, C₂ = cos } where x of coordinates matrix P. C+B P = C←B [f(x)] c = [ The coordinates of a function f(x) relative to the basis B are [f(x)] B = [8] 6 coordinates of f(x) relative to C and find f(x). f(x) = Ex: 3 Ex: 3 nn 2 sin x + -, ne Z nn 2,n € Z is also a basis for V. Find the change COS x Find the
Verify that this is a vector set check each property(assoctive, etc): X = (x0, x#) Y = (yo, y#) X + Y (xo + yo, x# + y#) scalar × X = X = (sclar × x0, sclar × x#) X
Let V = R?, defined the addition by: %3D (u1 +u2) + (V1 + v2)= (U1 +v1 , Uz +V2 + 1) With standard scalar multiplication. Is V a vector space justify ?
Using linear algebra, show that G={p1(x)=1−x+x^2, p2(x)=x, p3(x)=x+x^2} is a basis of P2, the vector space of polynomials of degree at most two with real coefficients.
Only part d please.
Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the magnitude of p(x)=2+1.7x equals: Answer:
Find the transpose, conjugate, and adjoint of (i) Transpose is idempotent: (4¹) = A. (ii) Transpose respects addition: (A + B) = A¹+ B¹. (iii) Transpose respects scalar multiplication: (c. A)¹ = c A. These three operations are defined even when m‡n. The transpose and adjoint are both functions from CX to Cxm These operations satisfy the following properties for all c E C and for all A, B € CX". (iv) Conjugate is idempotent: A = A. (v) Conjugate respects addition: A 6-3i 0 1 B = A + B (vi) Conjugate respects scalar multiplication: C. A = c. A. (vii) Adjoint is idempotent: (4†)t = 4. (viii) Adjoint respects addition: (A + B) =A+B+. (ix) Adjoint relates to scalar multiplication: (CA)t = c · At 2 + 12i 5+2.1i 2+5i -19i 17 3-4.5i
Identify all the polynomial functions of two variables of degree < 2 which are harmonic. Show that they form a vector space of dimension 5 and give a vector basis of that space.
Let V be a three-dimensional space of geometric vectors, and let a be a nonzero vector. Which of the following prescriptions defines a linear mapping A:V→V?
(b) (c) 0 (69) (89-19) 0 0 0 0 (6 9.89.163) 0 0 0 0 2 1 0 0 2 4. Determine whether the following vectors are lin- early independent in 2×2: 0 (@) [9] [8