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Consider the two linear mapsT1 and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3) and T2(x1, x2, x3) = (x1, 0,0)
- a)T is idempotent but T2 is not idempotent
- b)T2 is idempotent but T1 is not idempotent
- c)Both T1 and T2 are idempotent
- d)Neither T1 norT2 are idempotent.
Correct answer is option 'C'. Can you explain this answer?
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Consider the two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) =...
Idempotent Linear Maps:
An idempotent linear map is a linear map T such that T(T(x)) = T(x) for all x in the vector space V. In other words, applying the map T twice is the same as applying it once.
T1(x1, x2, x3) = (0, x2, x3):
Let's calculate T1(T1(x1, x2, x3)) and see if it is equal to T1(x1, x2, x3).
Applying T1 once:
T1(x1, x2, x3) = (0, x2, x3)
Applying T1 twice:
T1(T1(x1, x2, x3)) = T1(0, x2, x3) = (0, x2, x3)
Since T1(T1(x1, x2, x3)) = T1(x1, x2, x3), we can see that T1 is idempotent.
T2(x1, x2, x3) = (x1, 0, 0):
Now let's calculate T2(T2(x1, x2, x3)) and see if it is equal to T2(x1, x2, x3).
Applying T2 once:
T2(x1, x2, x3) = (x1, 0, 0)
Applying T2 twice:
T2(T2(x1, x2, x3)) = T2(x1, 0, 0) = (x1, 0, 0)
Since T2(T2(x1, x2, x3)) ≠ T2(x1, x2, x3), we can see that T2 is not idempotent.
Conclusion:
So, from our calculations, we can conclude that:
- T1 is idempotent as T1(T1(x)) = T1(x) for all x in V.
- T2 is not idempotent as T2(T2(x)) ≠ T2(x) for all x in V.
Hence, the correct answer is option 'C': Both T1 and T2 are idempotent.
An idempotent linear map is a linear map T such that T(T(x)) = T(x) for all x in the vector space V. In other words, applying the map T twice is the same as applying it once.
T1(x1, x2, x3) = (0, x2, x3):
Let's calculate T1(T1(x1, x2, x3)) and see if it is equal to T1(x1, x2, x3).
Applying T1 once:
T1(x1, x2, x3) = (0, x2, x3)
Applying T1 twice:
T1(T1(x1, x2, x3)) = T1(0, x2, x3) = (0, x2, x3)
Since T1(T1(x1, x2, x3)) = T1(x1, x2, x3), we can see that T1 is idempotent.
T2(x1, x2, x3) = (x1, 0, 0):
Now let's calculate T2(T2(x1, x2, x3)) and see if it is equal to T2(x1, x2, x3).
Applying T2 once:
T2(x1, x2, x3) = (x1, 0, 0)
Applying T2 twice:
T2(T2(x1, x2, x3)) = T2(x1, 0, 0) = (x1, 0, 0)
Since T2(T2(x1, x2, x3)) ≠ T2(x1, x2, x3), we can see that T2 is not idempotent.
Conclusion:
So, from our calculations, we can conclude that:
- T1 is idempotent as T1(T1(x)) = T1(x) for all x in V.
- T2 is not idempotent as T2(T2(x)) ≠ T2(x) for all x in V.
Hence, the correct answer is option 'C': Both T1 and T2 are idempotent.
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Consider the two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. Can you explain this answer?
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Consider the two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. 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 two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. 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 two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. Can you explain this answer?.
Consider the two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. 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 two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. 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 two linear mapsT1and T2 on V3 defined as T1(x1, x2, x3) = (0, x2, x3)and T2(x1, x2, x3) = (x1, 0,0)a)T is idempotent but T2 is not idempotentb)T2 is idempotent but T1 is not idempotentc)Both T1and T2 are idempotentd)Neither T1 norT2 are idempotent.Correct answer is option 'C'. Can you explain this answer?.
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