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(a) When F = R, determine the number of equivalent classes of symmetric bilinear forms on V. (This is a translation of exercise 26, which is explained in teacher's lecture. Still, you need to give out all details.) (b) When F = C, determine the number of equivalent classes of symmetric bilinear forms on V. (Hint. First show that every symmetric bilinear form on V has a matrix representation using only non-negative numbers as entries.) Definition A.1. A bilinear form H on V is called skew-symmetric if H(x, y) = -H(y, x) for all x, y ≤ V. (c) Let F be a field with char F #2. Use the following theorem A.2 to determine the number of equivalent classes of skew-symmetric bilinear forms on V.

(a) When F = R, determine the number of equivalent classes of symmetric bilinear forms on V. (This is a translation of exercise 26, which is explained in teacher's lecture. Still, you need to give out all details.) (b) When F = C, determine the number of equivalent classes of symmetric bilinear forms on V. (Hint. First show that every symmetric bilinear form on V has a matrix representation using only non-negative numbers as entries.) Definition A.1. A bilinear form H on V is called skew-symmetric if H(x, y) = -H(y, x) for all x, y ≤ V. (c) Let F be a field with char F #2. Use the following theorem A.2 to determine the number of equivalent classes of skew-symmetric bilinear forms on V.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Let V be a finite-dimensional vector space over a field F, dim V = n and H₁, H₂ be two bilinear forms on V. We say H₁ and H₂ are equivalent on V if there exists bases B1, B₂ of V such that 3₁ (H₁) = 3₂ (H₂). Under this definition, the set B(V) of all bilinear forms on V is partitioned into several equivalent classes. (a) When F = R, determine the number of equivalent classes of symmetric bilinear forms on V. (This is a translation of exercise 26, which is explained in teacher's lecture. Still, you need to give out all details.) (b) When F = C, determine the number of equivalent classes of symmetric bilinear forms on V. (Hint. First show that every symmetric bilinear form on V has a matrix representation using only non-negative numbers as entries.) Definition A.1. A bilinear form H on V is called skew-symmetric if H(x, y) = − H(y, x) for all x, y ≤ V. (c) Let F be a field with char F 2. Use the following theorem A.2 to determine the number of equivalent classes of skew-symmetric bilinear forms on V. Theorem A.2. Let F be a field with char F ‡ 2, V be a finite dimensional vector space over F, dim V = n, and H be a skew-symmetric bilinear form on V. Then there exists a basis of V such that the matrix representation (H) takes the form Ok - Ik (n-2k)×k Ik Ok 0(n-2k)xk Okx(n-2k) Okx(n-2k) On-2k
Transcribed Image Text:2. Let V be a finite-dimensional vector space over a field F, dim V = n and H₁, H₂ be two bilinear forms on V. We say H₁ and H₂ are equivalent on V if there exists bases B1, B₂ of V such that 3₁ (H₁) = 3₂ (H₂). Under this definition, the set B(V) of all bilinear forms on V is partitioned into several equivalent classes. (a) When F = R, determine the number of equivalent classes of symmetric bilinear forms on V. (This is a translation of exercise 26, which is explained in teacher's lecture. Still, you need to give out all details.) (b) When F = C, determine the number of equivalent classes of symmetric bilinear forms on V. (Hint. First show that every symmetric bilinear form on V has a matrix representation using only non-negative numbers as entries.) Definition A.1. A bilinear form H on V is called skew-symmetric if H(x, y) = − H(y, x) for all x, y ≤ V. (c) Let F be a field with char F 2. Use the following theorem A.2 to determine the number of equivalent classes of skew-symmetric bilinear forms on V. Theorem A.2. Let F be a field with char F ‡ 2, V be a finite dimensional vector space over F, dim V = n, and H be a skew-symmetric bilinear form on V. Then there exists a basis of V such that the matrix representation (H) takes the form Ok - Ik (n-2k)×k Ik Ok 0(n-2k)xk Okx(n-2k) Okx(n-2k) On-2k
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