The span of some vectors consists of all linear combinations of these vectors. A linear combination of vectors is just a finite sum of scalars times vectors.
Definition 18.9.1 Let

That is, it is a sum of scalars times the vectors for some choice of scalars a_{1},
Observation 18.9.2 Let

consists of all vectors which are of the form

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

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
Definition 18.9.3 The vectors Ax where x ∈ F^{p} is also called the column space of A and also Im
What do you really mean when you say there is a solution x to a linear system of equations Ax = b? You mean that b is in the span of the columns of A. After all, if A =
A subspace is a set of vectors with the property that linear combinations of these vectors remain in the set. Geometrically, subspaces are like lines and planes which contain the origin. More precisely, the following definition is the right way to think of this.
Definition 18.9.4 Let V be a nonempty collection of vectors in F^{n}. Then V is called a subspace if whenever α,β are scalars and u,v are vectors in V, the linear combination αu + βv is also in V .
There is no substitute for the above definition or equivalent algebraic definition! However, it is sometimes helpful to look at pictures at least initially. The following are four subsets of ℝ^{2}. 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, multiplication of a vector in the set by a nonnegative scalar results in a vector in the set as does the sum of two vectors in the set. However, multiplication by a negative scalar does not take a vector in the set to another in the set.
Observe how the above definition indicates that the claims posted on the picture are valid. Now here are the two main examples of subspaces.
Theorem 18.9.5 Let A be an m×n matrix. Then Im

Then ker
Proof: Suppose Ax_{i} is in Im

this because of the above properties of matrix multiplication. Note that A0 = 0 so 0 ∈ Im
Now suppose x,y are both in N

Thus the condition is satisfied. Of course N
Subspaces are exactly those subsets of F^{n} which are themselves vector spaces. Recall that a vector space is something which satisfies the vector space axioms on Page 807.
Proposition 18.9.6 Let V be a nonempty collection of vectors in F^{n}. Then V is a subspace if and only if V is itself a vector space having the same operations as those defined on F^{n}.
Proof: Suppose first that V is a subspace. It is obvious all the algebraic laws hold on V because it is a subset of F^{n} and they hold on F^{n}. Thus u + v = v + u along with the other axioms. Does V contain 0? Yes because it contains 0u = 0. Are the operations defined on V ? That is, when you add vectors of V do you get a vector in V ? When you multiply a vector in V by a scalar, do you get a vector in V ? Yes. This is contained in the definition. Does every vector in V have an additive inverse? Yes because −v =
Next suppose V is a vector space. Then by definition, it is closed with respect to linear combinations. Hence it is a subspace. ■