Linear Algebra

84 Chapter 2. Vector Spaces
A vector space must have at least one element, its zero vector. Thus a
one-element vector space is the smallest one possible.
1.7 Definition A one-element vector space is a trivial space.
Warning! The examples so far involve sets of column vectors with the usual
operations. But vector spaces need not be collections of column vectors, or even
of row vectors. Below are some other types of vector spaces. The term ‘vector
space’ does not mean ‘collection of columns of reals’. It means something more
like ‘collection in which any linear combination is sensible’.
1.8 Example Consider P3 = {a0 + a1x + a2x 2 + a3x 3 � � a0,... ,a3 ∈ R}, the
set of polynomials of degree three or less (in this book, we’ll take constant
polynomials, including the zero polynomial, to be of degree zero). It is a vector
space under the operations
(a0 + a1x + a2x 2 + a3x 3 )+(b0 + b1x + b2x 2 + b3x 3 )
=(a0 + b0)+(a1 + b1)x +(a2 + b2)x 2 +(a3 + b3)x 3
and
r · (a0 + a1x + a2x 2 + a3x 3 )=(ra0)+(ra1)x +(ra2)x 2 +(ra3)x 3
(the verification is easy). This vector space is worthy of attention because these
are the polynomial operations familiar from high school algebra. For instance,
3 · (1 − 2x +3x 2 − 4x 3 ) − 2 · (2 − 3x + x 2 − (1/2)x 3 )=−1+7x 2 − 11x 3 .
Although this space is not a subset of any R n , there is a sense in which we
can think of P3 as “the same” as R 4 . If we identify these two spaces’s elements
in this way
a0 + a1x + a2x 2 + a3x 3
corresponds to
then the operations also correspond. Here is an example of corresponding additions.
1 − 2x +0x2 +1x3 + 2+3x +7x2 − 4x3 3+1x +7x2 − 3x3 ⎛ ⎞
1
⎜
corresponds to ⎜−2
⎟
⎝ 0 ⎠
1
+
⎛ ⎞
2
⎜ 3 ⎟
⎝ 7 ⎠
−4
=
⎛ ⎞
3
⎜ 1 ⎟
⎝ 7 ⎠
−3
Things we are thinking of as “the same” add to “the same” sum. Chapter Three
makes precise this idea of vector space correspondence. For now we shall just
leave it as an intuition.
⎛
⎜
⎝
a0
a1
a2
a3
⎞
⎟
⎠