Page 202 - Matrix Analysis & Applied Linear Algebra

4.4 Basis and Dimension 197
Example 4.4.2

• If Z = {0} is the trivial subspace, then dim Z = 0 because the basis for
this space is the empty set.

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• If L is a line through the origin in , then dim L = 1 because a basis for
L consists of any nonzero vector lying along L.
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• If P is a plane through the origin in , then dim P = 2 because a minimal
spanning set for P must contain two vectors from P.
1 0 0

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• dim = 3 because the three unit vectors 0 , 1 , 0 constitute
0 0 1
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a basis for .
n n
• dim = n because the unit vectors {e 1 , e 2 ,..., e n } in form a basis.
Example 4.4.3
Problem: If V is an n -dimensional space, explain why every independent
subset S = {v 1 , v 2 ,..., v n }⊂V containing n vectors must be a basis for V.

Solution: dim V = n means that every subset of V that contains more than n
vectors must be linearly dependent. Consequently, S is a maximal independent
subset of V, and hence S is a basis for V.

Example 4.4.2 shows that in a loose sense the dimension of a space is a
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measure of the amount of “stuff” in the space—a plane P in has more
“stuff” in it than a line L, but P contains less “stuff” than the entire space
3 n
. Recall from the discussion in §4.1 that subspaces of are generalized
versions of flat surfaces through the origin. The concept of dimension gives us a
way to distinguish between these “flat” objects according to how much “stuff”
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they contain—much the same way we distinguish between lines and planes in .
Another way to think about dimension is in terms of “degrees of freedom.” In
the trivial space Z, there are no degrees of freedom—you can move nowhere—
whereas on a line there is one degree of freedom—length; in a plane there are
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two degrees of freedom—length and width; in there are three degrees of
freedom—length, width, and height; etc.
It is important not to confuse the dimension of a vector space V with the
number of components contained in the individual vectors from V. For example,
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if P is a plane through the origin in , then dim P =2, but the individual
vectors in P each have three components. Although the dimension of a space V
and the number of components contained in the individual vectors from V need
not be the same, they are nevertheless related. For example, if V is a subspace of
n
, then (4.3.16) insures that no linearly independent subset in V can contain
more than n vectors and, consequently, dim V≤ n. This observation generalizes
to produce the following theorem.