Let U be a supspace in V Let V1 Vk be a basis of V Let W1

Let U be a supspace in V, Let {V_1, ..., V_k} be a basis of V Let {W_1, ..., W_s} elementof V. Then prove that {[w_1, ..., [w_s]]} is linearly independent in V/V if and only if {V_1, V_k, W_1, ....W_s} is linearly independent in V.

Solution

Definition of Linearly independent vector is

v₁ , v₂----v_n are LI vectors if and only if   

the eqn a₁v₁+ a₂v₂+-----a_nv_n=0 => all a_i  =0------(1)

v₁ , v₂---v_k form the basis of the subspace U ( hence these k vectors are LI) ----(2)

the above k vectors and some more LI vectors form the basis of V

w₁, w₂ -----w_s are LI in V/U if and only if the k vectors in \'2\' and these \'s vectors form the basis of V

ie v₁ ,v₂ ---v_n , w₁ , w₂ ---w_n form the basis of the vector space V and hence these vectors ae LI