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Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.?
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Let Mn(R) be the real vector space of all n × n matrices with real ent...
Introduction:
We are given a real vector space Mn(R) consisting of all n × n matrices with real entries, where n is greater than or equal to 2. We need to consider the subspace W of Mn(R) spanned by the set {In, A, A^2, ...}, where A is a given matrix in Mn(R).
Definition of the subspace W:
The subspace W is defined as the set of all linear combinations of the matrices In, A, A^2, A^3, ..., where In denotes the identity matrix of order n.
Dimension of a vector space:
The dimension of a vector space is defined as the number of linearly independent vectors that span the space.
Proof:
To determine the dimension of the subspace W, we need to find the maximum number of linearly independent matrices that can be obtained from the set {In, A, A^2, ...}.
Case 1: A is a diagonalizable matrix:
If A is diagonalizable, it can be written as A = PDP^(-1), where D is a diagonal matrix and P is an invertible matrix. In this case, we can express any power of A as A^k = PD^kP^(-1), where D^k is a diagonal matrix obtained by raising each diagonal entry of D to the power k.
Since D is a diagonal matrix, the powers of D are simply obtained by raising each diagonal entry to the power k. Therefore, the matrices In, A, A^2, ..., A^n-1 are linearly independent.
Hence, in this case, the dimension of W is n.
Case 2: A is not a diagonalizable matrix:
If A is not diagonalizable, it means that A has fewer than n linearly independent eigenvectors. In this case, the powers of A will have linear dependencies.
Since the maximum number of linearly independent matrices that can be obtained from the set {In, A, A^2, ...} is n, the dimension of W is at most n.
Conclusion:
The dimension of the subspace W over R is necessarily at most n, as proven in the above cases. Therefore, the correct answer is option (D) at most n.
We are given a real vector space Mn(R) consisting of all n × n matrices with real entries, where n is greater than or equal to 2. We need to consider the subspace W of Mn(R) spanned by the set {In, A, A^2, ...}, where A is a given matrix in Mn(R).
Definition of the subspace W:
The subspace W is defined as the set of all linear combinations of the matrices In, A, A^2, A^3, ..., where In denotes the identity matrix of order n.
Dimension of a vector space:
The dimension of a vector space is defined as the number of linearly independent vectors that span the space.
Proof:
To determine the dimension of the subspace W, we need to find the maximum number of linearly independent matrices that can be obtained from the set {In, A, A^2, ...}.
Case 1: A is a diagonalizable matrix:
If A is diagonalizable, it can be written as A = PDP^(-1), where D is a diagonal matrix and P is an invertible matrix. In this case, we can express any power of A as A^k = PD^kP^(-1), where D^k is a diagonal matrix obtained by raising each diagonal entry of D to the power k.
Since D is a diagonal matrix, the powers of D are simply obtained by raising each diagonal entry to the power k. Therefore, the matrices In, A, A^2, ..., A^n-1 are linearly independent.
Hence, in this case, the dimension of W is n.
Case 2: A is not a diagonalizable matrix:
If A is not diagonalizable, it means that A has fewer than n linearly independent eigenvectors. In this case, the powers of A will have linear dependencies.
Since the maximum number of linearly independent matrices that can be obtained from the set {In, A, A^2, ...} is n, the dimension of W is at most n.
Conclusion:
The dimension of the subspace W over R is necessarily at most n, as proven in the above cases. Therefore, the correct answer is option (D) at most n.
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Let Mn(R) be the real vector space of all n × n matrices with real ent...
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Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.?
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Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.?.
Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.?.
Solutions for Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? in English & in Hindi are available as part of our courses for Mathematics.
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Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.?, a detailed solution for Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? has been provided alongside types of Let Mn(R) be the real vector space of all n × n matrices with real entries, n ≥ 2. Let A ∈ Mn(R). Consider the subspace W of Mn(R) spanned by {In, A, A2 , . . .}. Then the dimension of W over R is necessarily (A) ∞. (B) n 2 . (C) n. (D) at most n.? theory, EduRev gives you an
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