Chapter 1.4 Definitions (Linear Combinations and Systems of Linear Equations)

All definitions from Chapter 1.4

Linear Combination

Let V be a vector space and S a nonempty subset of V.

A vector v in V is called a ______ ________ of vectors of S if there exist a finite number of vectors u1, u2, …, un in S and scalars a1, a2, …, an in F such that

v=a1u1 + a2u2 + … + anun

Coefficients of Linear Combination

a1, a2, …, an in F

Span of S, or span(S)

The ____ __ _ is the set consisting of all linear combinations of the vectors in S.

span{∅}=0

Example:

span of set {(1,0,0),(0,1,0)} consists of all vectors in R³ that have the form a(1,0,0)+b(0,1,0)=(a,b,0) for scalars a,b in F.

Theorem 1.5

The span of any subset S of a vector space V is a subspace of V that contains S. Moreover, any subspace of V that contains S must also contain the span of S.

Generates/Spans

A subset S of a vector space V ________ (or _____) V if span(S)=V

We can also say that the vectors of S ________ (or ____) V.