# Spanning Sets

Definition. Let V be a vector space over a field F, and let S be a subset of V. The span of S is

That is, the span consists of all linear combinations of vectors in S.

S spans a subspace W of V if ; that is, if every element of W is a linear combination of elements of S.

Example. Let

(a) Prove or disprove: is in the span of S.

(b) Prove or disprove: is in the span of S.

(a) I try to find numbers a and b such that

(Remember that by default, when you convert a vector in "parenthesis form" like " " to a matrix, you convert the vector to a column vector.)

This is equivalent to the matrix equation

Row reduce the augmented matrix:

The solution is , . That is,

The vector is in the span of S.

(b) I try to find numbers a and b such that

This is equivalent to the matrix equation

Row reduce to solve the system:

The last matrix says " ", a contradiction. The system is inconsistent, so there are no such numbers a and b. Therefore, is not in the span of S.

Thus, to determine whether the vector is in the span of , , ..., in , form the augmented matrix

If the system has a solution, b is in the span, and coefficients of a linear combination of the v's which add up to b are given by a solution to the system. If the system has no solutions, then b is not in the span of the v's.

(In a general vector space where vectors may not be "numbers in slots", you have to got back to the definition of spanning set.)

Example. Consider the following set of vectors in :

Prove that the span of the set is all of .

Let . I must find real numbers a, b, c such that

The matrix equation is

Row reduce to solve:

I get the ugly solution

This shows that, given any vector , I can find a linear combination of the original three vectors which equals .

Thus, the span of the original set of three vectors is all of .

Example. Determine whether the vector is in the span of the set

(a) .

(b) .

(a) I want to find a and b such that

This is the system

Form the augmented matrix and row reduce:

The last matrix says and . Therefore, is in the span of S:

(b) I want to find a and b such that

This is the system

Form the augmented matrix and row reduce:

The last row of the row reduced echelon matrix says " ". This contradiction implies that the system is has no solutions. Therefore, is not in the span of S.

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