Teaching Haryana (HPSC) Assistant Professor Mock Test 2025 Mathematical Science Linear Algebra Linear Dependence, Basis & Dimension
On the complex vector space ℂ100 with standard basis {e1, e2, ..., e100}, consider the bilinear form B(x, y) = ∑ixiyi, where xi and yi are the coefficients of ei in x and y respectively. Which of the following statements are true?
1
B is nondegenerate.
2
Restriction of B to all nonzero subspaces is nondegenerate.
3
There is a 51 dimensional subspace W of ℂ100 such that the restriction B : W × W → ℂ is the zero map.
4
There is a 49 dimensional subspace W of ℂ100 such that the restriction B : W × W → ℂ is the zero map.
5
Question Not Attempted