The Big Basis Theorem

Theorem:  Let V be a vector space which is the span of finitely many vectors.  [Such V's are called finite dimensional.]

Then:

(1) V has a finite basis; that is, V has a basis with finitely many vectors. 
In fact, if V = Span {v₁, ...,v_k}, then some subset of these n vectors forms a basis for V.

(2) Every basis for V has the same number of elements.  This common number is called the dimension of V and is denoted by dim V. 

For the rest of the statement of this theorem, let n = dim V.

(3) [The Structure Theorem] If B={v₁, ...v_n} is a basis for V, then the the function T: V ---> Rⁿ given by T(v) = [v]_B is an isomorphism; that is, T is linear, one-to-one, and onto.s

(4) More than n vectors in V are dependent.

(5) Less than n vectors in V cannot span V.

(6) If k vectors span V, then k > n.

(7) If k vectors are independent, then k < n.

(8) Any collection of independent vectors can be enlarged to be a basis.

(9) Any collection of spanning vectors can be shrunk to be a basis. (This is just (1) above, again.)

(10) Any collection of n independent vectors automatically span, and is thus a basis.

(11) Any collection of n spanning vectors is automatically independent, and is thus a basis.

(12) If H is a subspace of V, then H is (also) finite dimensional and dim H < dim V.  Furthermore, dim H = dim V if and only if H = V.