T: P₂→ P₂ is a linearoperator. Determine whether the set\{T(1), T(x), T(x ^ 2)\} is a basis for P2. 19. . T(a * x ^ 2 + bx + c) = (a + b + c) * x ^ 2 + (a + b) * x + a
Question
T: P₂→ P₂ is a linearoperator. Determine whether the set{T(1), T(x), T(x ^ 2)} is a basis for P2.
- . T(a * x ^ 2 + bx + c) = (a + b + c) * x ^ 2 + (a + b) * x + a
Solution
To determine whether the set {T(1), T(x), T(x^2)} is a basis for P2, we first need to find the images of 1, x, and x^2 under the transformation T.
Given T(a * x^2 + b*x + c) = (a + b + c) * x^2 + (a + b) * x + a, we can find:
T(1) = T(0 * x^2 + 0x + 1) = (0 + 0 + 1) * x^2 + (0 + 0) * x + 0 = x^2 T(x) = T(0 * x^2 + 1x + 0) = (0 + 1 + 0) * x^2 + (0 + 1) * x + 0 = x^2 + x T(x^2) = T(1 * x^2 + 0*x + 0) = (1 + 0 + 0) * x^2 + (1 + 0) * x + 1 = x^2 + x + 1
So, the set {T(1), T(x), T(x^2)} = {x^2, x^2 + x, x^2 + x + 1}.
A set of vectors is a basis for a vector space if it is linearly independent and spans the vector space. In this case, the set {x^2, x^2 + x, x^2 + x + 1} is not linearly independent because the third vector can be expressed as a linear combination of the first two vectors (x^2 + x + 1 = 1*(x^2) + 1*(x^2 + x)). Therefore, the set {T(1), T(x), T(x^2)} is not a basis for P2.
Similar Questions
33. Let T: P₂→ P₂ be defined by T(ax² + bx + c) = ax^2 + (a-2b)x + b a. Determine whether p(x) = 2x^2 - 4x +6 is in the range of T. b. Find a basis for R(T).
In each part, determine whether the given vector p(1) in P₂ belongs to span (p₁(t), p₂(t), p3(1)), where p_{1}(t) = t ^ 2 + 2t + 1 p_{2}(t) = t ^ 2 + 3 and p_{1}(t) = t - 1 (a) p(t) = t ^ 2 + t + 2 (b) p(t) = 2t ^ 2 + 2t + 3 (c) p(t) = - t ^ 2 + t - 4 (d) p(t) = - 2t ^ 2 + 3t + 1
show that the transformation defined by T(x1,x2)=(2x1-3x2,x1+4,5x2) is not linear ChatGPT
If T : Rn → Rn is given by T (x1, x2, ..., xn) = (x2 + x3, x3, ..., xn, 0) then write downthe matrix of T w.r.t. the standard basis of Rn
[5marks] letT:C3→C3bealinearmapsuchthat T(v1) = 2v1, T(v2) = v1 + 2v2, T(v3) = v2 + 2v3. A) Find the matrix T , i.e., the matrix of the linear map T with respect to B,B the basis B in the domain and the basis B in the co-domain. B) Find Te2 . B C) Find Te2.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.