Linear Algebra/Span of a set

Definition
Let V be a vector space over a field F. Choose n vectors x1, x2, x3, ..., xn from the vector space V. The linear manifold spanned by x1, x2, x3, ..., xn is defined to be all elements of V of the form a1x1+a2x2+a3x3+...+anxn where a1, a2, a3, ..., an are all elements of the field F, and shall be denoted S(x1, x2, x3, ..., xn). This is obviously a linear subspace of the vector space V. Since every linear subspace of V contains x1, x2, x3, ..., xn and their linear combinations, S(x1, x2, x3, ..., xn) is the smallest subspace containing x1, x2, x3, ..., xn.

Theorem
If y1, y2, xy, ..., ym are elements of S(x1, x2, x3, ..., xn), then S(y1, y2, y3, ..., ym) is contained within S(x1, x2, x3, ..., xn)

Proof
All linear combinations of vectors which belong to a linear manifold also belong in the linear manifold (since a linear combination of linear combinations of vectors is also a linear combination of those vectors), and since any element of S(y1, y2, y3, ..., ym) is a linear combinations of vectors within the manifold, it too is within the set, thus proving that S(y1, y2, y3, ..., ym) is contained within S(y1, y2, y3, ..., ym).

Theorem
If x is linearly dependent on other vectors upon other vectors x1, x2, x3, ..., xn, then it belongs to S(x, x1, x2, x3, ..., xn) also

Proof
x, x1, x2, x3, ..., xn all belong to S(x1, x2, x3, ..., xn), then S(x, x1, x2, x3, ..., xn) must be contained within S(x, x1, x2, x3, ..., xn). Therefore, if x is linearly dependent upon x1, x2, x3, ..., xn, then S(x, x1, x2, x3, ..., xn) is equal to S(x1, x2, x3, ..., xn).

Theorem
The maximum number of linearly independent vectors of a set of vectors is equal to the dimension of the span of the set.

Proof
Suppose that there are d linearly independent vectors among x1, x2, x3, ..., xn with all other vectors being a linear combination of those d linearly independent vectors. This number d is the maximum number of linearly independent vectors among x1, x2, x3, ..., xn. Then any element of S(x1, x2, x3, ..., xn) must be a linear combination of those d linearly independent vectors, so they form a basis, and so d is the dimension of S(x1, x2, x3, ..., xn), which is equal to the maximum number of linearly independent vectors among x1, x2, x3, ..., xn.