This question concerns the following exercise.ExerciseConsider the linear transformationt: R³ → R³(x, y, z) → (2x, y + z, z-x).(i) Write down the matrix A of t with respect to the standard basisfor R3 for both the domain and codomain.(ii) Determine the matrix B of t with respect to thebasis {(1, 0, 1), (0,-1,-1), (1,1,0)} for both the domainand codomain.(iii) State whether there exists a vector v in R³ with the two propertiesVE Kert and ve Imt,and give a brief reason for your answer.(a) Explain why the following solution to this exercise is incorrect and/orincomplete, identifying one error or significant omission in each ofparts (i)-(iii). For each error or omission, explain the mistake thatthe writer of the solution has made.(There may be more than one error or omission in each part, but youneed identify only one. It should not be a statement or omission thatfollows logically from an earlier error or omission.)Solution (incorrect and/or incomplete!)20 -101 00 1(i) A =(ii) We have t(1, 0, 1) = (2, 1, 0), t(0, −1, −1) = (0, -2, -1)and t(1, 1,0) = (2, 1,-1), soB =0 2121 -20 -1 -1(iii) There is no such vector v. This is because Kert is a subset ofthe domain of t and Imt is a subset of the codomain of t.(b) Write out a correct solution to the exercise.

t: ℝ3→ℝ3x,y,z↦2x,y+z,z-x

Consider the linear transformation t: R³ R³ (x, y, z) → (2x, y + z, z-x). (i) Write down the matrix A of t with respect to the standard basis for R³ for both the domain and codomain. (ii) Determine the matrix B of t with respect to the basis {(1, 0, 1), (0, -1,-1), (1, 1,0)} for both the domain and codomain. (iii) State whether there exists a vector v in R³ with the two properties ve Kert and v € Imt, and give a brief reason for your answer.

Consider the linear transformation t: R³ R³ (x, y, z) → (2x, y + z, z-x). (i) Write down the matrix A of t with respect to the standard basis for R³ for both the domain and codomain. (ii) Determine the matrix B of t with respect to the basis {(1, 0, 1), (0, -1,-1), (1, 1,0)} for both the domain and codomain. (iii) State whether there exists a vector v in R³ with the two properties ve Kert and v € Imt, and give a brief reason for your answer.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 32E

See similar textbooks