Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Linear Algebra Bilinear Forms,Quadratic Forms
Let ℝ be the field of real numbers. Let V be the vector space of real polynomials of degree at most 1. Consider the bilinear form
〈 , 〉 : V × V → ℝ,
given by
\(\displaystyle \langle f, g\rangle=\int_0^1 f(x) g(x) d x\)
Which of the following is true?
1
For all nonzero real numbers a, b, there exists a real number c such that the vectors ax + b, x + c ∈ V are orthogonal to each other.
2
For all nonzero real numbers b, there are infinitely many real numbers c such that the vectors x + b, x + c ∈ V are orthogonal to each other.
3
For all positive real numbers c, there exist infinitely many real numbers a, b such that the vectors ax + b, x + c ∈ V are orthogonal to each other.
4
For all nonzero real numbers b, there are infinitely many real numbers c such that the vectors b, x + c ∈ V are orthogonal to each other.