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If T be a linear operator on a vector space V such that T2 - T + 1 = 0 then
- a)T is one-one but may not be onto.
- b)T is onto but may not be one-one.
- c)T is invertible.
- d)No such T exists.
Correct answer is option 'C'. Can you explain this answer?
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If T be a linear operator on a vector space V such that T2 - T + 1 = 0...
To prove that the correct answer is option 'C', we need to show that if T is a linear operator on a vector space V such that T^2 - T + 1 = 0, then T is invertible.
Given: T^2 - T + 1 = 0
To prove: T is invertible
Proof:
1. Assume that T is not invertible. This means that there exists a nonzero vector v in V such that T(v) = 0.
2. Applying T to both sides of the given equation, we get:
T(T^2 - T + 1) = T(0)
T^3 - T^2 + T = 0
3. Rearranging the terms, we have:
T(T^2 - T) = T^2
4. Since T(v) = 0, we can substitute T(v) with 0 in the above equation:
T(0) = T^2
0 = T^2
5. Multiplying both sides by T, we get:
T(0) = T(T^2)
0 = T^3
6. Substituting T^3 with T^2 - T in the above equation (from step 2), we have:
0 = T^2 - T
T^2 = T
7. Substituting T^2 with T in the given equation, we get:
T - T + 1 = 0
1 = 0
8. This is a contradiction, as 1 cannot be equal to 0.
9. Therefore, our assumption that T is not invertible is incorrect, and T must be invertible.
Hence, the correct answer is option 'C': T is invertible.
Given: T^2 - T + 1 = 0
To prove: T is invertible
Proof:
1. Assume that T is not invertible. This means that there exists a nonzero vector v in V such that T(v) = 0.
2. Applying T to both sides of the given equation, we get:
T(T^2 - T + 1) = T(0)
T^3 - T^2 + T = 0
3. Rearranging the terms, we have:
T(T^2 - T) = T^2
4. Since T(v) = 0, we can substitute T(v) with 0 in the above equation:
T(0) = T^2
0 = T^2
5. Multiplying both sides by T, we get:
T(0) = T(T^2)
0 = T^3
6. Substituting T^3 with T^2 - T in the above equation (from step 2), we have:
0 = T^2 - T
T^2 = T
7. Substituting T^2 with T in the given equation, we get:
T - T + 1 = 0
1 = 0
8. This is a contradiction, as 1 cannot be equal to 0.
9. Therefore, our assumption that T is not invertible is incorrect, and T must be invertible.
Hence, the correct answer is option 'C': T is invertible.
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