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Let V and W are two vector spaces over a field F.
Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?
Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?
- a)If ST is one-to-one, then S is one-to-one
- b)If V = W and V is finite-dimensional such that is — I, then T is invertible
- c)If dim V = 2 and dim W = 3, ST is invertible
- d)If TS is onto then S is onto
Correct answer is option 'A,C,D'. Can you explain this answer?
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Most Upvoted Answer
Let V and W are two vector spaces over a field F.Let S : V → W an...
(a) ST is one-one then T will be one - one not S.
(b) TS = I ⇒ TS is one-one and onto linear transformation
⇒ S is one-one and T is onto.
Now in this case
S : V → V and T : V → V
and T is onto
⇒ dim (R(T)) = dim (V)
⇒ dim (N(T)) = 0
(b) TS = I ⇒ TS is one-one and onto linear transformation
⇒ S is one-one and T is onto.
Now in this case
S : V → V and T : V → V
and T is onto
⇒ dim (R(T)) = dim (V)
⇒ dim (N(T)) = 0
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Let V and W are two vector spaces over a field F.Let S : V → W an...
X W be a bilinear map, i.e. a function that is linear in each variable separately.
One important property of bilinear maps is that they induce a linear map from the tensor product of V and W to some other vector space. This induced map is denoted by T : V ⊗ W -> X, where X is some other vector space.
The induced map T is defined as follows: for any v ∈ V and w ∈ W, we have T(v ⊗ w) = S(v, w).
This induced map T is linear in each variable separately, i.e. for any v, v' ∈ V and w, w' ∈ W, we have T((v + v') ⊗ w) = T(v ⊗ w) + T(v' ⊗ w) and T(v ⊗ (w + w')) = T(v ⊗ w) + T(v ⊗ w').
The induced map T is unique in the sense that any other linear map from V ⊗ W to X that agrees with S on elementary tensors must be equal to T.
In summary, given a bilinear map S : V x W -> X, there exists a unique induced linear map T : V ⊗ W -> X such that T(v ⊗ w) = S(v, w) for all v ∈ V and w ∈ W.
One important property of bilinear maps is that they induce a linear map from the tensor product of V and W to some other vector space. This induced map is denoted by T : V ⊗ W -> X, where X is some other vector space.
The induced map T is defined as follows: for any v ∈ V and w ∈ W, we have T(v ⊗ w) = S(v, w).
This induced map T is linear in each variable separately, i.e. for any v, v' ∈ V and w, w' ∈ W, we have T((v + v') ⊗ w) = T(v ⊗ w) + T(v' ⊗ w) and T(v ⊗ (w + w')) = T(v ⊗ w) + T(v ⊗ w').
The induced map T is unique in the sense that any other linear map from V ⊗ W to X that agrees with S on elementary tensors must be equal to T.
In summary, given a bilinear map S : V x W -> X, there exists a unique induced linear map T : V ⊗ W -> X such that T(v ⊗ w) = S(v, w) for all v ∈ V and w ∈ W.
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Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer?
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Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer?.
Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let V and W are two vector spaces over a field F.Let S : V → W and T : W → V be linear transformations. Then which of the following’s is/are not true?a)If ST is one-to-one, then S is one-to-oneb)If V = W and V is finite-dimensional such that is — I, then T is invertiblec)If dim V = 2 and dim W = 3, ST is invertibled)If TS is onto then S is ontoCorrect answer is option 'A,C,D'. Can you explain this answer?.
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