文档介绍：Chapter Five
More Dimensions
The Space R n
We are now prepared to move on to spaces of dimension greater than three. These
spaces are a straightforward generalization of our Euclidean space of three dimensions. Let
n be a positive integer. The n-dimensional Euclidean space R n is simply the set of
1
all ordered n-tuples of real numbers x = (x1 , x2 ,K,xn ) . Thus R is simply the real
numbers, R 2 is the plane, and R 3 is Euclidean three-space. These ordered n-tuples are
called points, or vectors. This definition does not contradict our previous definition of a
vector in case n =3 in that we identified each vector with an ordered triple (x1 , x2 , x3 ) and
spoke of the triple as being a vector.
We now define various arithmetic operations on R n in the obvious way. If we
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have vectors x = (x1 , x2 ,K,xn ) and y = (y1 , y2 ,K, yn ) in R , the sum x + y is defined
by
x + y = (x1 + y1 , x2 + y2 ,K, xn + yn ) ,
and multiplication of the vector x by a scalar a is defined by
ax = (ax1, ax2 ,K,axn ) .
It is easy to verify that a(x + y) = ax + ay and (a + b)x = ax + bx .
Again we see that these definitions are entirely consistent with what we have done
in dimensions 1, 2, and 3-there is nothing to unlearn. Continuing, we define the length,
or norm of a vector x in the obvious manner
2 2 2
| x|= x1 + x2 +K+xn .
The scalar product of x and y is