Let V be a real vector space and v1 v2 vn be a basis for V

Let V be a real vector space and {v_1, v_2, ..., v_n} be a basis for V. Prove that S = {v_1, v_2, ... v_n, 0} is linearly dependent. (0 is the zero vector in V)

Solution

V is given here a real vector space .

Now its subspace S contains 0 ,here we have to prove

A subspace is closed under the operations of the vector space it is in .In this case if you add two vectors in the space ,their sum must be in it.So ,if you take any vector in the space and add its negative vector ,its sum is the zero vector ,means zero vector must be in the subspace,which is then definition in the subspace .

There is a condition for linear dependency ,that there must be a non zero vector present in it .

0 vector is sum of two vectors ,one negative and one positive ,it simply iplies that there are atleast two vectors which are non zero.Hence prove that S is linearly dependent.