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Let V be the space of twice differentiable functions on R satisfying f'' - 2f' + f = 0. Define T : V → R2 by T(f) = (f'(0), f(0)), Then T is

  • a)
    one - one and onto

  • b)
    one - one but not onto

  • c)
    onto but not one - one

  • d)
    neither one - one nor onto

Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Let V be the space of twice differentiable functions on R satisfying f...








⇒ T cannot be onto

⇒ T not one-one
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Most Upvoted Answer
Let V be the space of twice differentiable functions on R satisfying f...
To define the operator T on V, we need to specify how it acts on each function in V.

Given a function f(x) in V, we have the equation f(x) - 2f'(x) + f''(x) = 0.

Thus, we can define T(f) as T(f)(x) = 2f'(x) - f''(x).

In other words, T(f) is the second derivative of f subtracted from twice the first derivative of f.
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Let V be the space of twice differentiable functions on R satisfying f - 2f + f = 0. Define T : V→R2 by T(f) = (f(0), f(0)), Then T isa)one - one and ontob)one - one but not ontoc)onto but not one - oned)neither one - one nor ontoCorrect answer is option 'D'. Can you explain this answer?
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