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(c) Consider the quadratic form f(x1, x2, x3) = 3(a₁x₁ + a2x2 + a3x3)² = 4(b₁x₁ + b₂x2 + b3x3)². (i) Let a = = a1 a2 03 В = b₁ b₂ b3 Show that the matrix of the quadratic form is 3aa¹ - 488¹. (ii) Show that if a and 3 are unit vectors and are orthogonal to each other, then there is an orthogonal change of variable 8-8 P so that f(x1, x2, x3) can be written as 3y² - 4y2.

(c) Consider the quadratic form f(x1, x2, x3) = 3(a₁x₁ + a2x2 + a3x3)² = 4(b₁x₁ + b₂x2 + b3x3)². (i) Let a = = a1 a2 03 В = b₁ b₂ b3 Show that the matrix of the quadratic form is 3aa¹ - 488¹. (ii) Show that if a and 3 are unit vectors and are orthogonal to each other, then there is an orthogonal change of variable 8-8 P so that f(x1, x2, x3) can be written as 3y² - 4y2.

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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(c) Consider the quadratic form f(x1, x2, x3) = 3(a₁x₁ + a2x2 + a3x3)² − 4(b₁x₁ + b₂x2 + b3x3)². (i) Let a = a1 a2 a3 В = b₁ b₂ b3 Show that the matrix of the quadratic form is 3aa¹ - 488¹. (ii) Show that if a and 3 are unit vectors and are orthogonal to each other, then there is an orthogonal change of variable 8-8 = so that f(x1, x2, x3) can be written as 3y² - 4y2.
Transcribed Image Text:(c) Consider the quadratic form f(x1, x2, x3) = 3(a₁x₁ + a2x2 + a3x3)² − 4(b₁x₁ + b₂x2 + b3x3)². (i) Let a = a1 a2 a3 В = b₁ b₂ b3 Show that the matrix of the quadratic form is 3aa¹ - 488¹. (ii) Show that if a and 3 are unit vectors and are orthogonal to each other, then there is an orthogonal change of variable 8-8 = so that f(x1, x2, x3) can be written as 3y² - 4y2.
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