For a, b ∈ , let
p(x, y) = a²x₁ y₁ + abx₂y₁+ abx₁y₂ + b²x₂y₂, x =(x₁, x₂), y = (y₁, y₂) ∈ ²
For what values of a and b does p ∶ ² × ² → define an inner product?

Question

Question

This question was previously asked in

CSIR-UGC (NET) Mathematical Science: Held on (26 Nov 2020)

Answer 1

Explanation:

∵ p(x, y) is bilinear transform. then matrix representation of p(x, y) is,

If p(x, y) is defining an inner product if A is positive definite i.e. all eigenvalues of A are positive.

but det (A) = 0 ⇒ all eigenvalues can't be positive.

(∵ at least one eigenvalue of A

⇒ A is not positive definite.

⇒ p(x, y) is not defining an inner product for any value of a, b.

Option (4) is true

Download Solution PDF