1. Prove whether or not the following are linear transformations. Make sure to provide a specific counter-example of a failed property when possible.(c) Let V = P2 and W = R2×2. Consider the transformation T: VW defined byT(ax² + bx + c) = ( caa-bc-2 c+3a¿).V = {f: R→R|f is integrable on [0, 1]}.(d) LetConsider transformation T: V→ R defined by|T(S) = ['s(f(x) dr.

Step 1:A transformation T:V→W is linear if it satisfies the following properties for all functions…

Answered: 1. Prove whether or not the following…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 5CM: Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

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Let T be a linear transformation from R2 into R2 such that T(4,2)=(2,2) and T(3,3)=(3,3). Find T(7,2).
Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.
Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.
Let T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
Let T:P2P4 be the linear transformation T(p)=x2p. Find the matrix for T relative to the bases B={1,x,x2} and B={1,x,x2,x3,x4}.
Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].
Let S={v1,v2,v3} be a set of linearly independent vectors in R3. Find a linear transformation T from R3 into R3 such that the set {T(v1),T(v2),T(v3)} is linearly dependent.
Show that the transformation T defined by T(x1, X2) = (2x,- 3x,, x1 +4, 5x,) is not linear. %3D
22. Let T : R² → R³ be a linear transformation such that T(x1, x2) = (x1 – 2x2, –x1 + 3x2, 3x1 – 2x2). Find x such that T(x) = (-1,4, 9).
Let f: R²R be defined by f((x, y)) = 8y - 4x + 1. Is ƒ a linear transformation? a. f((x1, y₁) + (x2, Y2)) (Enter x₁ as x1, etc.) f((x₁, y₁)) + f((x2, y₂)) = + Does f((x1, y₁) + (x2, 2)) = f((x₁, y₁)) + f((x2, ₂)) for all (x1, y₁), (x2, Y₂) € R²? choose b. f(c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all CER and all (x, y) = R²? choose c. Is f a linear transformation? choose
Let T: R² →R² be a linear transformation such that T (×₁,×₂) = (x₁ +×₂, 6x₁ +5x2). Find x such that T(x) = (5,32). X =

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