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Consider the following linear transformation from the vector space R2 into the vector space R3.
T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)
Then, the rank and nullity of T are respectively.
T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)
Then, the rank and nullity of T are respectively.
- a)2 and 0
- b)1 and 0
- c)1 and 1
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the following linear transformation from the vector sp...
We are given that a linear transformation
T : R2 —> R3 defined by
T(x, y) = (-x -y, 3x + 8y, 9x - 11y)
We need to find the rank and nullity of T.
Let x, y ∈ ker T.
Then
T(x, y) = (0, 0, 0)
Using the definition of T, we get
Comparing the components on both sides, we get
Solving for * and y, we get x = 0, y = 0.
Hence,ker T= {(0, 0)},
Therefore,Nullity of T = dim ker T = 0.
Using Rank Nullity theorem, we get Rank of T = 2 - Nullity of T
= 2 - 0 = 2 .
Hence, Rank and Nullity of T are 2 and 0 respectively.
T : R2 —> R3 defined by
T(x, y) = (-x -y, 3x + 8y, 9x - 11y)
We need to find the rank and nullity of T.
Let x, y ∈ ker T.
Then
T(x, y) = (0, 0, 0)
Using the definition of T, we get
Comparing the components on both sides, we getSolving for * and y, we get x = 0, y = 0.Hence,ker T= {(0, 0)},
Therefore,Nullity of T = dim ker T = 0.
Using Rank Nullity theorem, we get Rank of T = 2 - Nullity of T
= 2 - 0 = 2 .
Hence, Rank and Nullity of T are 2 and 0 respectively.
Most Upvoted Answer
Consider the following linear transformation from the vector sp...
We are given that a linear transformation
T : R2 —> R3 defined by
T(x, y) = (-x -y, 3x + 8y, 9x - 11y)
We need to find the rank and nullity of T.
Let x, y ∈ ker T.
Then
T(x, y) = (0, 0, 0)
Using the definition of T, we get
Comparing the components on both sides, we get
Solving for * and y, we get x = 0, y = 0.
Hence,ker T= {(0, 0)},
Therefore,Nullity of T = dim ker T = 0.
Using Rank Nullity theorem, we get Rank of T = 2 - Nullity of T
= 2 - 0 = 2 .
Hence, Rank and Nullity of T are 2 and 0 respectively.
T : R2 —> R3 defined by
T(x, y) = (-x -y, 3x + 8y, 9x - 11y)
We need to find the rank and nullity of T.
Let x, y ∈ ker T.
Then
T(x, y) = (0, 0, 0)
Using the definition of T, we get
Comparing the components on both sides, we getSolving for * and y, we get x = 0, y = 0.Hence,ker T= {(0, 0)},
Therefore,Nullity of T = dim ker T = 0.
Using Rank Nullity theorem, we get Rank of T = 2 - Nullity of T
= 2 - 0 = 2 .
Hence, Rank and Nullity of T are 2 and 0 respectively.
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Community Answer
Consider the following linear transformation from the vector sp...
Rank and Nullity of the Linear Transformation:
Rank of T:
- The rank of a linear transformation is the dimension of the image of the transformation.
- To find the rank of T, we need to determine the dimension of the column space of the transformation matrix.
- The transformation matrix for T is
[ -1 -1 ]
[ 3 8 ]
[ 9 -11 ]
- By performing row operations on the matrix, we can see that the rows are linearly independent.
- Therefore, the rank of T is equal to the number of non-zero rows in the matrix, which is 2.
Nullity of T:
- The nullity of a linear transformation is the dimension of the kernel (null space) of the transformation.
- To find the nullity of T, we need to determine the dimension of the null space of the transformation matrix.
- The null space is the set of all vectors x such that T(x) = 0.
- By solving the system of equations T(x) = 0, we find that the only solution is x = 0.
- This means that the null space of T contains only the zero vector, and hence the nullity of T is 0.
Therefore, the rank of T is 2 and the nullity of T is 0. Hence, the correct answer is option 'A'.
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Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. Can you explain this answer?
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Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. 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 Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. 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 Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. Can you explain this answer? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. Can you explain this answer?.
Solutions for Consider the following linear transformation from the vector space R2 into the vector space R3.T(x, y ) = (- x - y ,3x + 8y, 9x - 11y)Then, the rank and nullity of T are respectively.a)2 and 0b)1 and 0c)1 and 1d)None of theseCorrect answer is option 'A'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics.
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