Let v1 v2 vm in V be linearly independent and assume that v

Let v₁, v₂, ..., v_m in V be linearly independent and assume that v₁, v₂, ..., v_m do not span V. Show that there exists v_m+1 in V such that {v₁, v₂, ..., v_m, v_m+1} is a linearly independent set.

Solution

Let v₁, v₂, ..., v_m in V be linearly independent and assume that v1, v2, ..., vm do not span V. Show that there exists vm+1 in V such that {v1, v2, ..., vm, vm+1} is a linearly independent set.

Let, v₁, v₂, ..., v_m in V, be linearly independent, Now, span{v₁, v₂, ..., v_m} contains all the linear combinations of v₁, v₂, ..., v_m. Thus, if the set S = { v₁, v₂, ..., v_m} does not span V, then there exists a vector in V, say v_m+1 which cannot be expressed as a linear combination of the vectors v₁, v₂, ..., v_m. Let us assume that the set S’ = { v₁, v₂, ..., v_m, v_m+1} is linearly dependent. Then there exist scalars a₁, a₂,…,a_m , a_m+1 , not all zero, such that a₁v₁+a₂ v₂+…+a_m v_m +a_m+1 v_m+1 = 0. Then, v_m+1 = -1/a_m+1 (a₁v₁+a₂ v₂+…+a_m v_m) i.e. v_m+1 can be expressed as a linear combination of the vectors v₁, v₂, ..., v_m. This is a contradiction. Hence { v1, v2, ..., vm, vm+1} is a linearly independent set.