Page 118 - 35Linear Algebra

118 Linear Transformations

dimension is the number of independent directions available. To figure out
the dimension of a vector space, I stand at the origin, and pick a direction.
If there are any vectors in my vector space that aren’t in that direction, then
I choose another direction that isn’t in the line determined by the direction I
chose. If there are any vectors in my vector space not in the plane determined
by the first two directions, then I choose one of them as my next direction. In
other words, I choose a collection of independent vectors in the vector space
(independent vectors are defined in Chapter 10). A minimal set of indepen-
dent vectors is called a basis (see Chapter 11 for the precise definition). The
number of vectors in my basis is the dimension of the vector space. Every
vector space has many bases, but all bases for a particular vector space have
the same number of vectors. Thus dimension is a well-defined concept.
The fact that every vector space (over R) has infinitely many bases is
actually very useful. Often a good choice of basis can reduce the time required
to run a calculation in dramatic ways!
In summary:

A basis is a set of vectors in terms of which it is possible to
uniquely express any other vector.

6.5 Review Problems

Reading problems 1 , 2
Linear? 3
Webwork:
Matrix × vector 4, 5
Linearity 6, 7

1. Show that the pair of conditions:

L(u + v) = L(u) + L(v)
(1)
L(cv) = cL(v)
(valid for all vectors u, v and any scalar c) is equivalent to the single
condition:
L(ru + sv) = rL(u) + sL(v) , (2)
(for all vectors u, v and any scalars r and s). Your answer should have
two parts. Show that (1) ⇒ (2), and then show that (2) ⇒ (1).

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