Page 164 - Matrix Analysis & Applied Linear Algebra

CHAPTER 4

Vector

Spaces

4.1 SPACES AND SUBSPACES

After matrix theory became established toward the end of the nineteenth century,
it was realized that many mathematical entities that were considered to be quite
different from matrices were in fact quite similar. For example, objects such as
2
3
points in the plane , points in 3-space , polynomials, continuous functions,
and differentiable functions (to name only a few) were recognized to satisfy the
same additive properties and scalar multiplication properties given in §3.2 for
matrices. Rather than studying each topic separately, it was reasoned that it
is more efficient and productive to study many topics at one time by studying
the common properties that they satisfy. This eventually led to the axiomatic
definition of a vector space.
A vector space involves four things—two sets V and F, and two algebraic
operations called vector addition and scalar multiplication.
• V is a nonempty set of objects called vectors. Although V can be quite
general, we will usually consider V to be a set of n-tuples or a set of matrices.
• F is a scalar field—for us F is either the field of real numbers or the
field C of complex numbers.

• Vector addition (denoted by x + y ) is an operation between elements of V.
• Scalar multiplication (denoted by αx ) is an operation between elements of
F and V.
The formal definition of a vector space stipulates how these four things relate
to each other. In essence, the requirements are that vector addition and scalar
multiplication must obey exactly the same properties given in §3.2 for matrices.