1. Determine if the following functions are linear transformations. a) T: R³ → R³, defined by T(x, y, z) = (1, x + y, y + z) b) T: P² → P², defined by T(ax² + bx + c) = 3ax² + 2c

Algebra and Trigonometry (6th Edition)
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ChapterP: Prerequisites: Fundamental Concepts Of Algebra
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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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1. Determine if the following functions are linear transformations.
a) T: R³ → R³, defined by T(x, y, z) = (1, x + y, y + z)
b) T : P² → P², defined by T(ax² + bx + c) = 3ax² + 2c
2. For each linear transformation, determine ker(T), range(T), nullity(T), and rank(T).
Transcribed Image Text:1. Determine if the following functions are linear transformations. a) T: R³ → R³, defined by T(x, y, z) = (1, x + y, y + z) b) T : P² → P², defined by T(ax² + bx + c) = 3ax² + 2c 2. For each linear transformation, determine ker(T), range(T), nullity(T), and rank(T).
Expert Solution
Step 1: Definition of linear transformation

A linear transformation is a function

 T: Rnrightwards arrowRn that satisfies the following properties: 

  1. T(x+y)= T(x)+T(y)
  2. T(ax)= aT(x) 
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