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Let F be any field and let T be a linear operator on F2 defined by T(a, b) = (a + b, a), then T-1(a, b) is equal to:
- a)(b ,a -b)
- b)(a - b ,b)
- c)(a, a + b)
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let F be any field and let T be a linear operator on F2 defined by T(a...
Let F be any field and let T be a linear operator on F2 defined by
T(a, b) = (a + b, a)
We need to find T-1(a, b).
Let (a, b) ∈ ker T.
Then T(a, b) = (0, 0)
Using the definition of linear transformation, we get
(a + b, a) = (0, 0)
Comparing the components of the coordinates, we get
a + b = 0, a = 0
Solving for α and β, we get a = 0, b = 0.
Therefore,ker T = {(0,0)}
Hence, T is one-one. Since T is a linear transformation on finite dimensional vector space F2. Therefore, T is onto. Hence, T is invertible linear transformation.
Let (x,y) be the image of (a,b) under T-1.
Then (x, y) = T-1(a, b) or equivalently T(x, y) = (a, b)
Using the definition of linear transformation,
we get (x+y,y)= (a,b)
Comparing the components of the coordinates, we get
x + y = a and y = b
Solving for x and y, we get x= a - b , y = b.
Therefore,T-1(a, b) = (a - b, b)
T(a, b) = (a + b, a)
We need to find T-1(a, b).
Let (a, b) ∈ ker T.
Then T(a, b) = (0, 0)
Using the definition of linear transformation, we get
(a + b, a) = (0, 0)
Comparing the components of the coordinates, we get
a + b = 0, a = 0
Solving for α and β, we get a = 0, b = 0.
Therefore,ker T = {(0,0)}
Hence, T is one-one. Since T is a linear transformation on finite dimensional vector space F2. Therefore, T is onto. Hence, T is invertible linear transformation.
Let (x,y) be the image of (a,b) under T-1.
Then (x, y) = T-1(a, b) or equivalently T(x, y) = (a, b)
Using the definition of linear transformation,
we get (x+y,y)= (a,b)
Comparing the components of the coordinates, we get
x + y = a and y = b
Solving for x and y, we get x= a - b , y = b.
Therefore,T-1(a, b) = (a - b, b)
Most Upvoted Answer
Let F be any field and let T be a linear operator on F2 defined by T(a...
To find the inverse of the linear operator T, we need to find a linear operator T^-1 such that T^-1(T(a, b)) = (a, b) for all (a, b) in F^2.
Let's try to find the inverse operator T^-1 by using the given definition of T(a, b) = (a b, a).
Finding the Inverse Operator T^-1:
We want to find T^-1(a, b) such that T(T^-1(a, b)) = (a, b) for all (a, b) in F^2.
Let's apply T to T^-1(a, b):
T(T^-1(a, b)) = T(T^-1(a, b)) = T(T^-1(a, b)) = (T^-1(a, b) T^-1(a), T^-1(a)).
We want this to be equal to (a, b). So, we have the following equations:
T^-1(a, b) T^-1(a) = a (1)
T^-1(a) = b (2)
Solving Equation (2) for T^-1(a):
From Equation (2), we have T^-1(a) = b.
Substituting this into Equation (1), we get:
T^-1(a, b) b = a.
Expanding the left side, we have:
T^-1(a, b) = a - b.
Therefore, the inverse operator T^-1 is given by T^-1(a, b) = (a - b, b).
Comparing with the options:
The correct answer is option B) (a - b, b), which matches our derived inverse operator T^-1(a, b).
Thus, the inverse of the linear operator T(a, b) = (a b, a) is T^-1(a, b) = (a - b, b), as given in option B.
Let's try to find the inverse operator T^-1 by using the given definition of T(a, b) = (a b, a).
Finding the Inverse Operator T^-1:
We want to find T^-1(a, b) such that T(T^-1(a, b)) = (a, b) for all (a, b) in F^2.
Let's apply T to T^-1(a, b):
T(T^-1(a, b)) = T(T^-1(a, b)) = T(T^-1(a, b)) = (T^-1(a, b) T^-1(a), T^-1(a)).
We want this to be equal to (a, b). So, we have the following equations:
T^-1(a, b) T^-1(a) = a (1)
T^-1(a) = b (2)
Solving Equation (2) for T^-1(a):
From Equation (2), we have T^-1(a) = b.
Substituting this into Equation (1), we get:
T^-1(a, b) b = a.
Expanding the left side, we have:
T^-1(a, b) = a - b.
Therefore, the inverse operator T^-1 is given by T^-1(a, b) = (a - b, b).
Comparing with the options:
The correct answer is option B) (a - b, b), which matches our derived inverse operator T^-1(a, b).
Thus, the inverse of the linear operator T(a, b) = (a b, a) is T^-1(a, b) = (a - b, b), as given in option B.
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Let F be any field and let T be a linear operator on F2 defined by T(a, b) = (a +b, a), then T-1(a, b) is equal to:a)(b ,a -b)b)(a - b ,b)c)(a, a + b)d)None of theseCorrect answer is option 'B'. Can you explain this answer?
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