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Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]
The dimension d of all vector space W satisfies.
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The dimension d of all vector space W satisfies.
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Most Upvoted Answer
Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0,...
Understanding the Vector Space W
W is defined as the set of matrices obtained by evaluating polynomials at the matrix B, where B is given as:
- B = [[0, 1, 0],
[0, 0, 1],
[1, 0, 0]]
This matrix is a permutation matrix that cycles the standard basis vectors in R^3.
Polynomial Evaluation
- A polynomial p(x) can be expressed generally as:
p(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_1 * x + a_0
- Evaluating p(B) means substituting B into the polynomial.
Properties of B
- The matrix B has distinct eigenvalues, specifically the cube roots of unity (1, ω, ω²), where ω = e^(2πi/3).
- B is diagonalizable, and we can find a basis of eigenvectors corresponding to these eigenvalues.
Dimension of W
- The polynomials can be expressed in terms of their action on the eigenvalues of B.
- The minimal polynomial of B is of degree 3 because it captures the behavior of the eigenvalues.
Basis for W
- The polynomials up to degree 2 are sufficient to generate all possible matrices p(B) due to the nature of polynomial evaluation on a matrix.
- Thus, we can find a basis for W consisting of {I, B, B^2}.
Conclusion: Dimension of W
- The dimension d of W is 3, corresponding to the basis {I, B, B^2}.
- Therefore, the statement that the dimension d of W satisfies (a) 4 is incorrect. The correct dimension is 3.
W is defined as the set of matrices obtained by evaluating polynomials at the matrix B, where B is given as:
- B = [[0, 1, 0],
[0, 0, 1],
[1, 0, 0]]
This matrix is a permutation matrix that cycles the standard basis vectors in R^3.
Polynomial Evaluation
- A polynomial p(x) can be expressed generally as:
p(x) = a_n * x^n + a_(n-1) * x^(n-1) + ... + a_1 * x + a_0
- Evaluating p(B) means substituting B into the polynomial.
Properties of B
- The matrix B has distinct eigenvalues, specifically the cube roots of unity (1, ω, ω²), where ω = e^(2πi/3).
- B is diagonalizable, and we can find a basis of eigenvectors corresponding to these eigenvalues.
Dimension of W
- The polynomials can be expressed in terms of their action on the eigenvalues of B.
- The minimal polynomial of B is of degree 3 because it captures the behavior of the eigenvalues.
Basis for W
- The polynomials up to degree 2 are sufficient to generate all possible matrices p(B) due to the nature of polynomial evaluation on a matrix.
- Thus, we can find a basis for W consisting of {I, B, B^2}.
Conclusion: Dimension of W
- The dimension d of W is 3, corresponding to the basis {I, B, B^2}.
- Therefore, the statement that the dimension d of W satisfies (a) 4 is incorrect. The correct dimension is 3.
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Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4
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Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4 for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4 covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4 .
Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4 for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4 covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let W={p(B)) : p is a polynomial with real coefficient} where B = [[0, 1, 0], [0, 0, 1], [1, 0, 0]]The dimension d of all vector space W satisfies.(a) 4 .
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