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If q=x₁²+4x₁x₂-2x₂², find new variables y₁ and y₂ such that q=λ₁y₁²+λ₂y₂² for some λ₁ and λ₂ in R.?
If you could please give as much detail is possible. I need to learn how to do other questions like this. Thank you!
2 Answers
- AdrianoLv 69 years agoFavorite Answer
Recall that (a + b)² = a² + 2ab + b².
Looking at the 4x₁x₂ term, this suggests that we should try a = x₁ and b = 2x₂.
Thus, let y₁ = x₁ + 2x₂ and λ₁ = 1.
Then λ₁y₁² = 1(x₁ + 2x₂)² = x₁² + 4x₁x₂ + 4x₂²
We almost have what we want, but we are off by -6x₂².
Thus, let y₂ = x₂ and λ₂ = -6. Then:
q
= λ₁y₁² + λ₂y₂²
= (1)(x₁ + 2x₂)² + (-6)(x₂)²
= (x₁² + 4x₁x₂ + 4x₂²) - 6x₂²
= x₁² + 4x₁x₂ - 2x₂²
as desired.
