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Consider the vector space R3 and the maps f g : R3 —> R3 defined by f ( x , y, z) = (x, | y |, z) and g(x, y, z) = (x + 1, y - 1, z). Then,
  • a)
    both f and g are linear
  • b)
    neither f nor g is linear
  • c)
    g is linear but not f
  • d)
    f is linear but not g
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Consider the vector space R3 and the maps f g : R3 —> R3 defi...
We are given that two linear transformations , g : R3 --> R3 defined by
F(x ,y ,z ) = (x ,|y|,z)
and g(x, y, z) = (x + 1, y - 1, z)
Let (0 ,1,2) and (0, - 1 , 2) be two vectors of R3 then ( 0 ,1 , 2) = (0 ,1 ,2 ) 
and (0,-1 ,2 ) = (0 ,1 ,2 )
therefore f(0,1 ,2) + f(0 ,-1, 2) = (0,1,2) + (0,1,2) = (0, 2, 4) 
but f [(0,1,2) + (0,-1,2)] = f(0 0,4) = (0 ,0 ,4)
Hence, f(0,1,2) + f(0, -1, 2) ≠ [(0 ,1, 2) + (0,- 1, 2)]
Thus , f is non linear
Next, g(0, 0, 0) = (1, -1, 0)
but(1,- 1, 0) ≠ (0, 0, 0)
hence, g is also non linear.
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Most Upvoted Answer
Consider the vector space R3 and the maps f g : R3 —> R3 defi...
To specify a map, we need to define what it does to each vector in the vector space R3. Let's define f and g as follows:

f: R3 -> R3
f(x, y, z) = (2x, -y, 3z)

g: R3 -> R3
g(x, y, z) = (x + y, y - z, 2z)

In other words, f doubles the x-coordinate, negates the y-coordinate, and triples the z-coordinate of any vector in R3. g adds the x-coordinate and y-coordinate, subtracts the y-coordinate from the z-coordinate, and doubles the z-coordinate of any vector in R3.
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Consider the vector space R3 and the maps f g : R3 —> R3 defi...
D
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Consider the vector space R3 and the maps f g : R3 —> R3 defined by f ( x , y, z) = (x, | y |, z) and g(x, y, z) = (x + 1, y - 1, z). Then,a)both f and g are linearb)neither fnor g is linearc)g is linear but not fd)f is linear but not gCorrect answer is option 'B'. Can you explain this answer?
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Consider the vector space R3 and the maps f g : R3 —> R3 defined by f ( x , y, z) = (x, | y |, z) and g(x, y, z) = (x + 1, y - 1, z). Then,a)both f and g are linearb)neither fnor g is linearc)g is linear but not fd)f is linear but not gCorrect answer is option 'B'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Consider the vector space R3 and the maps f g : R3 —> R3 defined by f ( x , y, z) = (x, | y |, z) and g(x, y, z) = (x + 1, y - 1, z). Then,a)both f and g are linearb)neither fnor g is linearc)g is linear but not fd)f is linear but not gCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the vector space R3 and the maps f g : R3 —> R3 defined by f ( x , y, z) = (x, | y |, z) and g(x, y, z) = (x + 1, y - 1, z). Then,a)both f and g are linearb)neither fnor g is linearc)g is linear but not fd)f is linear but not gCorrect answer is option 'B'. Can you explain this answer?.
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