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Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?
  • a)
    nullity of T = nullity of S
  • b)
    dimension of U ≠ dimension of W
  • c)
    If dimension of V = 3, dimension of U = 4, then P is not identically zero
  • d)
    If dimension of V = 4, dimension of U = 3 and T is one- one, then P is identically zero
Correct answer is option 'C'. Can you explain this answer?
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Let U, V and W be finite dimensional real vector spaces, T : U → ...
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Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?a)nullity of T = nullity of Sb)dimension of U ≠ dimension of Wc)If dimension of V = 3, dimension of U = 4, then P is not identically zerod)If dimension of V = 4, dimension of U = 3 and T is one- one, then P is identically zeroCorrect answer is option 'C'. Can you explain this answer?
Question Description
Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?a)nullity of T = nullity of Sb)dimension of U ≠ dimension of Wc)If dimension of V = 3, dimension of U = 4, then P is not identically zerod)If dimension of V = 4, dimension of U = 3 and T is one- one, then P is identically zeroCorrect answer is option 'C'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?a)nullity of T = nullity of Sb)dimension of U ≠ dimension of Wc)If dimension of V = 3, dimension of U = 4, then P is not identically zerod)If dimension of V = 4, dimension of U = 3 and T is one- one, then P is identically zeroCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let U, V and W be finite dimensional real vector spaces, T : U → V, S : V → W and P : W → U be linear transformations. If range (ST) = nullspace (P), nullspace (ST) = range (P) and rank (T) = rank (S), then which one of the following is true?a)nullity of T = nullity of Sb)dimension of U ≠ dimension of Wc)If dimension of V = 3, dimension of U = 4, then P is not identically zerod)If dimension of V = 4, dimension of U = 3 and T is one- one, then P is identically zeroCorrect answer is option 'C'. Can you explain this answer?.
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