3: Consider the linear map T: R³ → P2 given byaT(6) = (a – 2b) + (3a + 4b)x + (–3a + 6b)x².a) Find the matrix of T using standard basis for R3 and basisB = {x²,x + x?, 1+x + x?} for P2.b) Find dim kerT and dim RangeT.

Let us consider the linear transformation, T: V→W Kernel of T is obtained as Ker(T)=v∈V|T(v)=0.…

Answered: 3: Consider the linear map T:R³ → P2…

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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5. Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x₁, x2, and x3 are not vectors but are entries of a vector x in R³. T(x1, x2, x3) = (2x1 — x2, x1 − 2x2 + x3, x1 − 3x2 + x3) - X1
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.
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a) Verify the given sequence B is a basis ; and b) compute the images of the standard basis vectors with respect to the given linear transformation. 15. fi = 1+2x – 2x2, f2 = 2 + 3x – 4x2, f3 = 1 + 4x – x², B = (f1, f2, f3). - 1 T(fi) = (; 10)- 4 T(fi) = (; ).T ,T(f2) : 3 8
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3: Consider the linear map T:R³ → P2 given by a T( ) = (a – 26) + (3a + 46)x + (-3a + 6b)x². a) Find the matrix of T using standard basis for R3 and basis B = {r², x + x²,1+x+ x²} for P2. b) Find dim kerT and dim RangeT.