Chapter 4 Vector Spaces And Modules
Up to this point we have been introduced to groups and to rings; the former has its motivation in the set of one-to-one mappings of a set onto itself, the latter, in the set of integers.
The third algebraic model which
Section 3.10 Polynomials Over The Rational Field
We specialize the general discussion to that of polynomials whose coecients are rational
numbers. Most of the time the coecients will actually be integers. For such polynomials we
shall be concerned with th
3.9 Polynomial Rings
Let F be a eld. By the ring of polynomials in the indeterminate, x, written as F [x], we
mean the set of all symbols a0 + a1 x + . . . + an xn , where n can be any nonnegative integer and
where the coecients a1 , a2 , . . . , an are a
Section 4.2 Linear Independence And Bases
If we look somewhat more closely at two of the examples described in the previous section,
namely Example 4.1.4 and Example 4.1.3, we notice that although they do have many properties
in common there is one striki
Section 4.4 Inner Product Spaces
In our discussion of vector spaces the specic nature of F as a eld, other than the fact that
it is a eld, has played virtually no role. In this section we no longer consider vector spaces V
over arbitrary elds F ; rather,
Chapter 5 Fields
In our discussion of rings we have already singled out a special class which we called elds.
A eld, let us recall, is a commutative ring with unit element in which every nonzero element
has a multiplicative inverse. Put another way, a eld
3.8 A Particular Euclidean Ring
Let J [i] = cfw_a + bi | a, b Z. We call these the Gaussian integers. Our rst objective is
to exhibit J [i] as a Euclidean ring. In order to do this we must rst introduce a function d(x)
dened for every nonzero element in J
Section 4.3 Dual Spaces
Given any two vector spaces, V and W , over a eld F, we have dened Hom(V, W ) to be
the set of all vector space homomorphisms of V into W . As yet Hom(V, W ) is merely a set
with no structure imposed on it. We shall now proceed to
F
x F
F [x]
f (x) = 0 xn + 1 xn1 + . . . + i xni + . . . + n1 x + n
F [x]
f (x), f (x)
f (x) = n0 xn1 + (n 1)1 xn2 + . . . + (n i)i xni1 + . . . + n1
F [x]
F
F ma = 0 a = 0 F m > 0,
Section 3.11 Polynomial Rings Over Commutative Rings
In dening the polynomial ring in one variable over a eld F, no essential use was made of
the fact that F was a eld; all that was used was that F was a commutative ring. The eld
nature of F only made its