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Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively?
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Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by ...
Rank and Nullity of T
To determine the rank and nullity of the linear map T, we need to analyze the matrix M.
Matrix M:
M = [[1, 2], [0, 0]]
Definition of T:
T(A) = AM
Vector Space:
V is the vector space of all 2 * 2 real matrices.
Rank of T:
The rank of T is the dimension of the range of T, which is the set of all possible outputs of T.
To find the rank of T, we need to determine the dimension of the range of T. The range of T is the set of all matrices that can be obtained by multiplying M with a matrix A from V.
Observations:
- The first column of M is [1, 0], which means that the first column of any matrix obtained by multiplying M with A will be a linear combination of the columns of A.
- The second column of M is [2, 0], which means that the second column of any matrix obtained by multiplying M with A will be twice the first column of A.
Case 1: Rank of T is 0
If the first column of A is [0, 0], then the first column of the resulting matrix will also be [0, 0]. This means that the rank of T is at least 1.
Case 2: Rank of T is 1
If the first column of A is not [0, 0], then the first column of the resulting matrix will be a linear combination of the columns of A. Since the second column of the resulting matrix will be twice the first column of A, it will always be a scalar multiple of the first column. Therefore, the rank of T is 1.
Nullity of T:
The nullity of T is the dimension of the null space of T, which is the set of all matrices A from V such that T(A) = 0.
Since the rank of T is 1, the nullity of T is the dimension of the set of all matrices A from V such that multiplying M with A results in the zero matrix.
Observations:
- The second column of M is [2, 0], which means that the second column of any matrix obtained by multiplying M with A will be twice the first column of A.
- The null space of T is the set of all matrices A from V such that the first column of A is [0, 0].
Nullity of T:
Since the null space of T consists of all matrices A from V such that the first column of A is [0, 0], the nullity of T is the dimension of this set, which is 2.
Summary:
- The rank of T is 1.
- The nullity of T is 2.
To determine the rank and nullity of the linear map T, we need to analyze the matrix M.
Matrix M:
M = [[1, 2], [0, 0]]
Definition of T:
T(A) = AM
Vector Space:
V is the vector space of all 2 * 2 real matrices.
Rank of T:
The rank of T is the dimension of the range of T, which is the set of all possible outputs of T.
To find the rank of T, we need to determine the dimension of the range of T. The range of T is the set of all matrices that can be obtained by multiplying M with a matrix A from V.
Observations:
- The first column of M is [1, 0], which means that the first column of any matrix obtained by multiplying M with A will be a linear combination of the columns of A.
- The second column of M is [2, 0], which means that the second column of any matrix obtained by multiplying M with A will be twice the first column of A.
Case 1: Rank of T is 0
If the first column of A is [0, 0], then the first column of the resulting matrix will also be [0, 0]. This means that the rank of T is at least 1.
Case 2: Rank of T is 1
If the first column of A is not [0, 0], then the first column of the resulting matrix will be a linear combination of the columns of A. Since the second column of the resulting matrix will be twice the first column of A, it will always be a scalar multiple of the first column. Therefore, the rank of T is 1.
Nullity of T:
The nullity of T is the dimension of the null space of T, which is the set of all matrices A from V such that T(A) = 0.
Since the rank of T is 1, the nullity of T is the dimension of the set of all matrices A from V such that multiplying M with A results in the zero matrix.
Observations:
- The second column of M is [2, 0], which means that the second column of any matrix obtained by multiplying M with A will be twice the first column of A.
- The null space of T is the set of all matrices A from V such that the first column of A is [0, 0].
Nullity of T:
Since the null space of T consists of all matrices A from V such that the first column of A is [0, 0], the nullity of T is the dimension of this set, which is 2.
Summary:
- The rank of T is 1.
- The nullity of T is 2.
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Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively?
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Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively?.
Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let M = [[1, 2], [0, 0]] and top: V -> V be the linear map defined by T(A)= AM where V be the vector space of all 2 * 2 real matrices. Then, rank and nullity of T respectively?.
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