3. Which of the following mappings are linear transformations? Give a proof (directly using thedefinition of a linear transformation) or a counterexample in each case. [Recall that Pn(F) is thevector space of all real polynomials p(x) of degree at most n with values in F.]·(2) = (3n+2)=) ·(i) 0 : R³ → R² given by 0 y3y zax4 + bx² + c).(ii) : P2(F) → P₁(F) given by (p(x)) = p(x²) (so (ax² + bx + c) = ax4þ

Part (i): θ:R3→R2 given by θ⎝⎛xyz⎠⎞=(z3y+z). Step 1: Check AdditivityLet u=⎝⎛x1y1z1⎠⎞ and…

Answered: 3. Which of the following mappings are…

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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Similar questions

A translation in R2 is a function of the form T(x,y)=(xh,yk), where at least one of the constants h and k is nonzero. (a) Show that a translation in R2 is not a linear transformation. (b) For the translation T(x,y)=(x2,y+1), determine the images of (0,0,),(2,1), and (5,4). (c) Show that a translation in R2 has no fixed points.
Let T be a linear transformation from R2 into R2 such that T(1,0)=(1,1) and T(0,1)=(1,1). Find T(1,4) and T(2,1).
Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.
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:R3R3 be the linear transformation that projects u onto v=(2,1,1). (a) Find the rank and nullity of T. (b) Find a basis for the kernel of 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 T be a linear transformation T such that T(v)=kv for v in Rn. Find the standard matrix for T.
For the linear transformation from Exercise 45, let =45 and find the preimage of v=(1,1). 45. Let T be a linear transformation from R2 into R2 such that T(x,y)=(xcosysin,xsin+ycos). Find a T(4,4) for =45, b T(4,4) for =30, and c T(5,0) for =120.
Let T1 : R? → R² and T2 : R? → R? be linear transformations defined as follows. H)- -2x1 T1 -5x1 + 4x2 1: (E)- -4x1 T2 -3x2 (G) (F)- Ex: 42 (T1 o T2) (D Ex: 42 (T2 o T1)
is this correct??
Determine which of the following functions are linear transformations. For each function, either prove that it is a linear transformation of explain why it is not. (a) S: R² R² given by (b) T: P₂ → P₁ given by I s([*]) = [2+³+1] T(co+c₁x + c₂²) = −c₁ + ₂x. where co, C1, C₂ are constants. (c) F: M2,2 M2,2 given by F(A) = A-AT.
х1 — 2х2 -x1 +3x2. Find * = Зх1 — 2х2 Let T : R2 → R be a linear transformation such that T( X1 X1 X2 x2] such that T(x): 9.

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