Im stuck on the following question:
Let V be a vector space. Prove that if it is possible to find m vectors in V which are linearly independent, and n vectors which span V, then m must be less than, or equal to, n.
Any help/hints would be great
Im stuck on the following question:
Let V be a vector space. Prove that if it is possible to find m vectors in V which are linearly independent, and n vectors which span V, then m must be less than, or equal to, n.
Any help/hints would be great
Call the linearly independent vectors $\displaystyle L={v_0, v_1...v_m}$ and the set of n vectors which spans V $\displaystyle S={u_0, u_2...u_n}$.
Assume m>n. Then there is at least one more vector in L than in S. L is a subset of V so must be in the span(S)=V. Since any vector in v is linearly independent, it cannot be expressed by a combination of other vectors in V, or of span(S). Even if all vectors in S are independent, the number of independent vectors in L is always greater than those in S, thus at least one vector in L is not in the span(S), which implies not in V, making a contradiction.
Kind of messy but that's the basic idea.