Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Linear Algebra Inner Product Spaces, Orthonormal Basis
Let V be the vector space of polynomials in the variable t of degree at most 2 over ℝ. An inner product on V is defined by
\(\langle f, g\rangle=\int_0^1 f(t) g(t) d t\)
for f, g ∈ V, Let W = span {1 - t2,1 + t2} and \(\mathrm{W}^{\perp} \) be the orthogonal complement of W in V. Which of the following conditions is satisfied for all h ∈ \(\mathrm{W}^{\perp} \)?
1
h is an even function, i.e. h(t) = h(-t)
2
h is an odd function, i.e. h(t) = -h(-t)
3
h(t) = 0 has a real solution
4
h(0) = 0
5
Question Not Attempted