**An Example of Creating a
Definition in**

**The Keys to Linear
Algebra**

**4.3.3 Linearly
Dependent and Independent Vectors**

For *n*-vectors, the property needed to
ensure a unique solution to (4.20) is that of vectors
"opening up properly.'' For example, the vectors **u**
and **v** in Figure 4.7(a) and in Figure 4.7(b) open
up properly. In contrast, the vectors **u **and **v
**in Figure 4.8(a) and in Figure 4.8(b) do not open up
properly. This is because the vectors in Figure 4.8(a) lie on top
of each other and the vectors in Figure 4.8 (b) point in exactly
opposite directions. The objective now is to create a
mathematical definition of what it means for vectors to
"open up properly'' by translating the associated visual
image to symbolic form.

One approach is to develop a definition for the property satisfied by the vectors in Figure 4.7. Alternatively, you can first develop a definition for the property of "not opening up properly,'' satisfied by the vectors in Figure 4.8, and then write the negation of that definition to capture the property of the vectors in Figure 4.7.

**Linearly Dependent
Vectors**

Following the latter approach, try to identify
similarities between the pair of vectors in Figure 4.8(a) and the
pair in Figure 4.8(b). For example, in Figure4.8(a), **u **points
in the same direction as **v **but has a different
length. Recall from Section 1.2.2 that you can change the length
of **v **(without affecting the direction) by
multiplying **v** by a nonnegative real number, say,
*a *>= 0. Thus, by an appropriate choice of the real number *a
*>= 0, you can make the vectors **u **and *a***v
**equal. You can translate this observation to the
following symbolic form using the quantifier *there is*:

There is a real number

a>= 0 such thatu=av(4.21)

Looking now at Figure 4.8(b), you will notice
that **u **points in the opposite direction to **v**.
In this case, by multiplying **v **by an appropriate
choice of the nonpositive real number *a *<= 0, you can
again make **u** and *a***v**
equal. This observation translates to the following symbolic
form:

There is a real number

a<= 0 such thatu=av(4.22)

Can you create a single statement that includes
both (4.21) and (4.22) as special cases? One such approach is to
allow *a* to be any real number---positive, negative, or
0. Thus, a unification of (4.21) and (4.22) is the following:

There is a real number

asuch thatu=av(4.23)

Generalizing the Definition to
3-Space. The next step is to generalize
(4.23) when the vectors lie in 3-space. Indeed, (4.23) is still
valid when **u **and **v** are vectors
in 3-space that lie on top of, or opposite to, each other.
However, what if you are working with a third vector, **w**,
in 3-space, such as the one in Figure 4.9? You might write the
following statement for this property:

There are real numbers

aandbsuch thatu=av +bw(4.24)

Is the property in (4.24) correct for all
groups of three vectors in 3-space that do not open up properly?
Only by extensive trials with other special cases of **u**,
**v**, and **w **will you discover that
(4.24) is not necessarily correct in all cases. For example, the
vectors in Figure 4.10 do not satisfy (4.24) because *a***v
+ ***b***w **always points in the same
direction as **v **and **w**. Rather,
the vectors in Figure 4.10 satisfy the following property:

There are real numbers

aandbsuch thatv=au +bw(4.25)

The vectors in Figure 4.10 also satisfy the property that

There are real numbers

aandbsuch thatw=au +bv(4.26)

In fact, at least one of (4.24), (4.25), or
(4.26) is always true for vectors **u**, **v**,
and **w** in 3-space that do not open up properly.
The challenge is to write a single statement that covers all
three of the special cases in (4.24), (4.25), and (4.26). Such a
unification requires cleverness by moving all vectors to the same
side of the equality sign and then realizing that there are real
numbers associated with each vector, that is,

There are real numbers

a,b, andcsuch thatau +bv +cw=0(4.27)

However, the real numbers in the special cases
are not just *any *real numbers ---they have some special
properties: in (4.24), *a* = 1; in (4.25), *b *= 1;
and in (4.26), *c* = 1. One way to capture this fact is to
require in (4.27) that at least one of the real numbers be 1. An
alternative, but equivalent, way to say this (as you are asked to
verify in Exercise 25) is the following:

There are real numbers

a,b, andc, not all 0, withau +bv +cw=0(4.28)

If a is not 0, for example, then statement
(4.28) is equivalent to saying that **u **is a
linear combination of **v **and **w**.

Generalizing the Definition to *n*-Space.
The next step is to generalize (4.28) to
the case where the vectors lie in *n*-space. So suppose
you have a group of *k* vectors in *n*-space that
do not open up properly. Once again, subscript and superscript
notation is helpful. Let each of **v**_{1},
..., **v**_{k }be an *n*-vector. By
introducing the term "linearly dependent'' for "not
opening up properly'' and by generalizing (4.28) in the natural
way, you obtain the following definition.

**Definition 4.6** The *n*-vectors
**v**_{1}, ..., **v**_{k}
are **linearly dependent **if and only if there are
real numbers *t*_{1}, ..., *t*_{k},
not all zero, such that

t_{1}v_{1}+^{... }+t_{k}v_{k }=0