Kenneth Kuttler

398 CHAPTER 18. LINEAR FUNCTIONS

Definition 18.4.1 Let{u1, · · · ,up

}be some vectors in Fn. A linear combination

of these vectors is a sum of the following form:

∑k=1

akuk

That is, it is a sum of scalars times the vectors for some choice of scalars a1, · · · ,ap.span(u1, · · · ,up) denotes the set of all linear combinations of these vectors.

Observation 18.4.2 Let{u1, · · · ,up

}be vectors in Fn. Form the n× p matrix A ≡(

u1 · · · up)

which has these vectors as columns. Then

span(u1, · · · ,up)

consists of all vectors which are of the form

Ax for x ∈ Fp.

Recall why this is so. A typical thing in what was just described is

(u1 · · · up

) x1...

= x1u1 + · · ·+ xnup

In other words, a typical vector of the form Ax is a linear combination of the columns of A.Thus we can write either span(u1, · · · ,up) or all Ax for x ∈ Fp to denote the same thing.

Definition 18.4.3 The vectors Ax where x ∈ Fp is also called the column space ofA and also Im(A) meaning image of A, also denoted as A(Fn). Thus column space equalsspan(u1, · · · ,up) where the ui are the columns of A.

As explained earlier, when you say there is a solution x to a linear system of equa-tions Ax= b, you mean that b is in the span of the columns of A. After all, if A =(u1 · · · up

), you are looking for x=

(x1 · · · xp

)T such that x1u1 + x2u2 +· · ·+ xpup = Ax= b.

A subspace is a set of vectors with the property that linear combinations of these vectorsremain in the set. Geometrically, subspaces are like lines and planes which contain theorigin. More precisely, the following definition is the right way to think of this.

Definition 18.4.4 Let V be a nonempty collection of vectors in Fn. Then V is calleda subspace if whenever α,β are scalars and u,v are vectors in V, the linear combinationαu+βv is also in V .

There is no substitute for the above definition or equivalent algebraic definition! How-ever, it is sometimes helpful to look at pictures at least initially. The following are foursubsets of R2. The first is the shaded area between two lines which intersect at the origin,the second is a line through the origin, the third is the union of two lines through the origin,and the last is the region between two rays from the origin. Note that in the last, multipli-cation of a vector in the set by a nonnegative scalar results in a vector in the set as does the

398 CHAPTER 18. LINEAR FUNCTIONSDefinition 18.4.1 Le: {uy ,Up} be some vectors in F". A linear combinationof these vectors is a sum of the following form:PVY agurgk=1That is, it is a sum of scalars times the vectors for some choice of scalars aj,--- dp.span (t1,--- ,t&,) denotes the set of all linear combinations of these vectors.Observation 18.4.2 Let {u1, ee up} be vectors in F". Form the n x p matrix A =( Ut oc: Up ) which has these vectors as columns. Thenspan (t1,--* , Up)consists of all vectors which are of the formAa forxz €F?.Recall why this is so. A typical thing in what was just described isx]( wy vee Up ) : SX1U +++ + Xp UpXpIn other words, a typical vector of the form Az is a linear combination of the columns of A.Thus we can write either span(uj,--- ,up) or all Ax for x € F? to denote the same thing.Definition 18.4.3 The vectors Ax where x € F? is also called the column space ofA and also Im (A) meaning image of A, also denoted as A(F"). Thus column space equalsspan (t1,--+ , Up) where the u; are the columns of A.As explained earlier, when you say there is a solution x to a linear system of equa-tions Ax = b, you mean that b is in the span of the columns of A. After all, if A =( Uj, oct Up ), you are looking for # = ( X] ttt Xp yt such that xjw) +x2u2 ++XpUp = Ax = Bb.A subspace is a set of vectors with the property that linear combinations of these vectorsremain in the set. Geometrically, subspaces are like lines and planes which contain theorigin. More precisely, the following definition is the right way to think of this.Definition 18.4.4 LerV bea nonempty collection of vectors in F”. Then V is calleda subspace if whenever a, B are scalars and u,v are vectors in V, the linear combinationau+ Bv is also in V.There is no substitute for the above definition or equivalent algebraic definition! How-ever, it is sometimes helpful to look at pictures at least initially. The following are foursubsets of R*. The first is the shaded area between two lines which intersect at the origin,the second is a line through the origin, the third is the union of two lines through the origin,and the last is the region between two rays from the origin. Note that in the last, multipli-cation of a vector in the set by a nonnegative scalar results in a vector in the set as does the