Suppose that v1 v2 v3 is a linearly independent subset of a

Suppose that {v_1, v_2, v_3} is a linearly independent subset of a vector space V with dlm(V) = 4 and that v_4 is not in span({i v_1, v_2, v_3}). Prove that (v_1, v_2, v_3, v_4) is a basis fof V.

Solution

Given that dim v =4

Hence there are 4 linearly independent vectors in the base of V

Already v1, v2, and v3 are linearly independent hence one cannot be represented as a linear combination of other 2.

Now consider v4 a vector in V which cannot be linearly represented as a combination of v1, v2, v3, and v4

Then the set S = {v1,v2,v3,v4} is a linearly independent set with dimension 4.

Since V is also of dimension 4, there cannot be more than 4 which can be added to the basis to make it linearly independent.

Hence S forms a basis for V

Thus proved.