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6.4 The Vector Space R n 165

We can write any n-vector in standard form
< x 1 , x 2 ,··· , x n >= x 1 e 1 + x 2 e 2 + ··· + x n e n .
This is a direct generalization of writing a 3-vector in terms of the orthonormal 3-vectors i, j
and k.

n
n
Suppose now that S is a set of vectors in R . We call S a subspace of R if the following
conditions are met:
1. O is in S.
2. The sum of any vectors in S is in S.
3. The product of any vector in S by any real number is in S.

Conditions (2) and (3) of this definition are equivalent to asserting that αF + βG is in S for
any numbers α and β and vectors F and G in S.
R is a subspace of itself, and the set S ={< 0,0,··· ,0 >} consisting of just the zero vector
n
n
is a subspace of R . This is called the trivial subspace. Here are more substantial examples.
EXAMPLE 6.8
2
n
Let S consist of all vectors in R having norm 1. In R this can be visualized as the set of points
on the unit circle about the origin, and in 3-space as the set of points on the unit sphere about the
n
origin. S is not a subspace of R because the zero vector is not in S, violating requirement (1) of
the definition. This is enough to disqualify S from being a subspace. However, in this example,
requirements (2) and (3) also fail. A sum of two vectors having length 1 does not have length
1, hence is not in S. And a scalar multiple of a vector in S is not in S unless the scalar is 1
or −1.

EXAMPLE 6.9
4
Let K consist of all scalar multiples of F =< −1,4,2,0 > in R . The zero vector is in K (this is
the product of F with the number zero). A sum of scalar multiples of F is a scalar multiple of F,
hence is in K, so requirement (2) holds. And a scalar multiple of a scalar multiple of F is also a
scalar multiple of F, so requirement (3) is true.

EXAMPLE 6.10

In R ,let W consist of all vectors having second, fourth and sixth component zero. Thus S con-
6
sists of all 6-vectors < x,0, y,0, z,0>. Then <0,0,0,0,0,0> is in W (choose x = y = z = 0).
A sum of vectors in W also has second, fourth and sixth components zero, as does any scalar
multiple of a vector in W. Therefore W is a subspace of R .
6

EXAMPLE 6.11
n
Let F 1 ,··· ,F k be any k vectors in R . Then the set L of all vectors of the form
α 1 F 1 + α 2 F 2 + ··· + α k F k ,
n
in which the α j s can be any real numbers, forms a subspace of R . We call this subspace the

span of F 1 ,··· ,F k and we will say more shortly about subspaces formed in this way.

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October 14, 2010 14:21 THM/NEIL Page-165 27410_06_ch06_p145-186