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Let (-, -) be a symmetric bilinear form on ℝ2 such that there exist nonzero v, w ∈ ℝ2 such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?
- a)A2 = 0.
- b)rank A = 1
- c)rank A = 0.
- d)there exists u ∈ ℝ2, u ≠ 0 such that (u, u) = 0.
Correct answer is option 'D'. Can you explain this answer?
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Explanation:
Symmetric bilinear form
- A symmetric bilinear form on a vector space V is a bilinear map B: V × V → R such that B(u, v) = B(v, u) for all u, v ∈ V.
Given Conditions
- We have a symmetric bilinear form (-, -) on 2.
- There exist nonzero vectors v, w ∈ 2 such that (v, v) > 0 and (w, w) < 0,="" and="" (v,="" w)="" />
Matrix Representation
- Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis.
- The entries of A are given by Aij = (ei, ej), where {ei} is the standard basis of 2.
Rank of A
- Since (v, w) = 0, the corresponding entry in the matrix A is zero.
- This implies that the rows of A are linearly dependent, leading to a rank deficiency.
- Therefore, the rank of A is less than 2, i.e., rank(A) = 1 or 0.
Existence of u
- Since (w, w) < 0,="" there="" exists="" a="" nonzero="" vector="" u="αw" for="" some="" α="" ∈="" r="" such="" that="" (u,="" u)="α^2(w," w)="" />
- Thus, there exists a vector u ∈ 2, u ≠ 0 such that (u, u) = 0.
Therefore, the correct statement is that there exists a vector u ∈ 2, u ≠ 0 such that (u, u) = 0, which corresponds to option 'd'.
Symmetric bilinear form
- A symmetric bilinear form on a vector space V is a bilinear map B: V × V → R such that B(u, v) = B(v, u) for all u, v ∈ V.
Given Conditions
- We have a symmetric bilinear form (-, -) on 2.
- There exist nonzero vectors v, w ∈ 2 such that (v, v) > 0 and (w, w) < 0,="" and="" (v,="" w)="" />
Matrix Representation
- Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis.
- The entries of A are given by Aij = (ei, ej), where {ei} is the standard basis of 2.
Rank of A
- Since (v, w) = 0, the corresponding entry in the matrix A is zero.
- This implies that the rows of A are linearly dependent, leading to a rank deficiency.
- Therefore, the rank of A is less than 2, i.e., rank(A) = 1 or 0.
Existence of u
- Since (w, w) < 0,="" there="" exists="" a="" nonzero="" vector="" u="αw" for="" some="" α="" ∈="" r="" such="" that="" (u,="" u)="α^2(w," w)="" />
- Thus, there exists a vector u ∈ 2, u ≠ 0 such that (u, u) = 0.
Therefore, the correct statement is that there exists a vector u ∈ 2, u ≠ 0 such that (u, u) = 0, which corresponds to option 'd'.
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Let (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option 'D'. Can you explain this answer?
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Let (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option '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 (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option '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 (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option 'D'. Can you explain this answer?.
Let (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option '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 (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option '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 (-, -) be a symmetric bilinear form on2such that there exist nonzero v, w ∈ 2such that (v, v) > 0 > (w, w) and (v, w) = 0. Let A be the 2 × 2 real symmetric matrix representing this bilinear form with respect to the standard basis. Which one of the following statements is true?a)A2= 0.b)rank A = 1c)rank A = 0.d)there exists u ∈ 2, u ≠ 0 such that (u, u) = 0.Correct answer is option 'D'. Can you explain this answer?.
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