Linear Algebra/Span of a set

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Definition[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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

Proof[edit]

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.