Page 120 - A Course in Linear Algebra with Applications

104 Chapter Four: Introduction to Vector Spaces

6. Prove that the vector spaces C[0, 1] and P(F) are infinitely
generated, where F is any field.
2
7. Let A and B be vectors in R . Show that A and B generate
2
R if and only if neither is a scalar multiple of the other.
Interpret this result geometrically.

4.3 Linear Independence in Vector Spaces

We begin with the crucial definition. Let V be a vector
space and let X be a non-empty subset of V. Then X is said to
be linearly dependent if there are distinct vectors Vi, v 2 ,..., v^
in X, and scalars c±, c 2 ,..., Ck, not all of them zero, such that

civi + c 2 v 2 H h c/jVfc = 0.

This amounts to saying that at least one of the vectors v$ can
be expressed as a linear combination of the others. Indeed, if
say Ci / 0, then we can solve the equation for v$, obtaining

n

For example, a one-element set {v} is linearly dependent if and
only if v = 0. A set with two elements is linearly dependent
if and only if one of the elements is a scalar multiple of the
other.
A subset which is not linearly dependent is said to be
linearly independent. Thus a set of distinct vectors {vi, .. ,
.
Vfc} is linearly independent if and only if an equation of the
form civi -I hCfcVfc = 0 always implies that c\ = c 2 = • • • =
= 0 .
c k
We shall often say that vectors v i , . . . , v& are linearly de-
pendent or independent, meaning that the subset {vi,..., v/J
has this property.