Page 169 - Matrix Analysis & Applied Linear Algebra

164 Chapter 4 Vector Spaces

any two vectors in P is also in P so that if u, v ∈P, then u + v ∈P. The
closure property for scalar multiplication (M1) holds because multiplying any
vector by a scalar merely stretches it, but its angular orientation does not change
3
so that if u ∈P, then αu ∈P for all scalars α. Lines and surfaces in that
have curvature cannot be subspaces for essentially the same reason depicted in
3
Figure 4.1.2. So the only proper subspaces of are the trivial subspace, lines
through the origin, and planes through the origin.
The concept of a subspace now has an obvious interpretation in the visual
3
2
spaces and —subspaces are the flat surfaces passing through the origin.
Flatness
Although we can’t use our eyes to see “flatness” in higher dimensions,
our minds can conceive it through the notion of a subspace. From now on,
think of flat surfaces passing through the origin whenever you encounter
the term “subspace.”

For a set of vectors S = {v 1 , v 2 ,..., v r } from a vector space V, the set of
all possible linear combinations of the v i ’s is denoted by
span (S)= {α 1 v 1 + α 2 v 2 + ··· + α r v r | α i ∈F} .
Notice that span (S) is a subspace of V because the two closure properties
ξ
(A1) and (M1) are satisfied. That is, if x = i i v i and y = i η i v i are two
linear combinations from span (S) , then the sum x + y = (ξ i + η i )v i is also
i
a linear combination in span (S) , and for any scalar β, βx = (βξ i )v i is
i
also a linear combination in span (S) .

αu + βv

βv

αu
v
u

Figure 4.1.4