Page 104 - A Course in Linear Algebra with Applications
P. 104
88 Chapter Four: Introduction to Vector Spaces
where the entries Xi are real numbers. Of course these are
special types of matrices, so rules of addition and scalar mul-
tiplication are at hand, namely
Xl
f \ /Vi
V2 X2 + V2
+
\x J V Vr>
n
and
fx \ ( CXi \_
x
X2 cx 2
Thus the set R n is "closed" with respect to the operation
of adding pairs of its elements, in the sense that one cannot
n n
escape from R by adding two of its elements; similarly R is
closed with respect to multiplication of its elements by scalars.
Notice also that R n contains the zero column vector.
Another point to observe is that the rules of matrix alge-
bra listed in 1.2.1 which are relevant to column vectors apply
n
n
to the elements of R . The set R , together with the op-
erations of addition and scalar multiplication, forms a vector
space which is known as n- dimensional Euclidean space.
Line segments and R 3
When n is 3 or less, the vector space R n has a good
3
geometrical interpretation. Consider the case of R . Atypical
element of R 3 is a 3-column
Assume that a cartesian coordinate system has been chosen
with assigned x, y and z -axes. We plan to represent the col-
umn vector A by a directed line segment in three-dimensional
