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Let T be linear transformation on R2 into itself such that T(1, 0) = (1, 2) and T(1, 1) = (0, 2). Then, T(a, b) is equal to
- a)(a, 2b)
- b)(2a, b)
- c)(a - b, 2a)
- d)(a - b, 2b)
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let T be linear transformation on R2 into itself such that T(1, 0) = (...
We are given that a linear transformation T : R2 —> R2 defined by
T(1,0) = T(1 ,2)
and T(1,1) = (0, 2)
We need to find the image of (a, b) under linear transformation T.
Let there exist α and β as scalars such that (a, b)= a (1, 0) + P(l, 1) or equivalently(a, b) = (a + β, β)
Implies a = α + β and b = β
Solving for α, β we get α = a - b
and β = b
Therefore,
(a, b) = ( a - b) (1, 0) + b(1, 1)
Taking the image under linear transformation T, we get
T(a, b) = (a - b) T(1, 0) + bT(1, 1) Implies T(a, b) = (a - b) (1, 2) + b(0, 2)
=(a- b, 2a)
T(1,0) = T(1 ,2)
and T(1,1) = (0, 2)
We need to find the image of (a, b) under linear transformation T.
Let there exist α and β as scalars such that (a, b)= a (1, 0) + P(l, 1) or equivalently(a, b) = (a + β, β)
Implies a = α + β and b = β
Solving for α, β we get α = a - b
and β = b
Therefore,
(a, b) = ( a - b) (1, 0) + b(1, 1)
Taking the image under linear transformation T, we get
T(a, b) = (a - b) T(1, 0) + bT(1, 1) Implies T(a, b) = (a - b) (1, 2) + b(0, 2)
=(a- b, 2a)
Most Upvoted Answer
Let T be linear transformation on R2 into itself such that T(1, 0) = (...
To determine the linear transformation T(a, b), we need to find a matrix representation of this transformation.
We are given that T(1, 0) = (1, 2) and T(1, 1) = (0, 2). We can write these equations as:
T(1, 0) = (a, b) = (1, 2)
T(1, 1) = (c, d) = (0, 2)
We can express T(a, b) as a linear combination of the standard unit vectors i and j:
T(a, b) = aT(1, 0) + bT(0, 1)
Let's find the values of T(1, 0) and T(0, 1) using the given information:
T(1, 0) = T(1 * (1, 1) - 0 * (1, 0)) = 1 * T(1, 1) - 0 * T(1, 0) = (0, 2) - (0, 0) = (0, 2)
T(0, 1) = T(0 * (1, 1) + 1 * (1, 0)) = 0 * T(1, 1) + 1 * T(1, 0) = (0, 0) + (1, 2) = (1, 2)
Now, we can express T(a, b) in terms of T(1, 0) and T(0, 1):
T(a, b) = aT(1, 0) + bT(0, 1) = a(0, 2) + b(1, 2) = (0, 2a) + (b, 2b) = (b, 2a+2b)
The transformation T(a, b) is equal to (b, 2a+2b), which matches option C: (a - b, 2a).
Therefore, the correct answer is option C: (a - b, 2a).
We are given that T(1, 0) = (1, 2) and T(1, 1) = (0, 2). We can write these equations as:
T(1, 0) = (a, b) = (1, 2)
T(1, 1) = (c, d) = (0, 2)
We can express T(a, b) as a linear combination of the standard unit vectors i and j:
T(a, b) = aT(1, 0) + bT(0, 1)
Let's find the values of T(1, 0) and T(0, 1) using the given information:
T(1, 0) = T(1 * (1, 1) - 0 * (1, 0)) = 1 * T(1, 1) - 0 * T(1, 0) = (0, 2) - (0, 0) = (0, 2)
T(0, 1) = T(0 * (1, 1) + 1 * (1, 0)) = 0 * T(1, 1) + 1 * T(1, 0) = (0, 0) + (1, 2) = (1, 2)
Now, we can express T(a, b) in terms of T(1, 0) and T(0, 1):
T(a, b) = aT(1, 0) + bT(0, 1) = a(0, 2) + b(1, 2) = (0, 2a) + (b, 2b) = (b, 2a+2b)
The transformation T(a, b) is equal to (b, 2a+2b), which matches option C: (a - b, 2a).
Therefore, the correct answer is option C: (a - b, 2a).
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Let T be linear transformation on R2 into itself such that T(1, 0) = (1, 2) and T(1, 1) = (0, 2). Then, T(a, b) is equal toa)(a, 2b)b)(2a, b)c)(a - b, 2a)d)(a - b, 2b)Correct answer is option 'C'. Can you explain this answer?
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