Question

Suppose $V$ is a finite dimensional vector space. For hyperplanes $$H_1, \ldots, H_r$$ in $V$ say they are linearly independent provided the corresponding linear subspaces in $$V^\vee$$ are linearly independent. Set $$n = \dim V$$ and show that $$H_1, \ldots, H_r$$ are linearly independent if and only if $$\dim( \cap H_i) = n - r$$ . (Hint: Write $$H_i = \ker \varphi_i$$ for $$\varphi_i \in V^\vee$$ and consider $$\Phi \colon V \to F^r$$ by $$\Phi(\mathbf{v}) = (\varphi_1(\mathbf{v}), \ldots, \varphi_r(\mathbf{v}))$$ ).

Answer 1

"H_i is are hyper plane is n -dimensional vector space V. these dim H_i = n -1 exist a basis B of H_i such that B = {b_1, . . .,b_m} we can extent this basis to B = {b_1,. . ., b_n -1, b_n} of V define phi_i: v rightarrow F by phi_i(b_n) = 1, phi_i(b_j) = 0 forall I elementsof {1, . . ., n -1} thus H_i = ker phi_i, phi_i elementsof V^2 consider the linear map counterintegral: v rightarrow F^r by counterintegral (V) = (phi_1(v)), . . ,phi_n(v)) then by rank-multity theorem, dim(ker(counterintegral)) + rank(counterintegral) = dim V . . .(1) but ker counterintegral = {v elementsof V