Page 110 - A Course in Linear Algebra with Applications

94 Chapter Four: Introduction to Vector Spaces

to be the set of all real polynomials in x of degree less than n.
Here we mean to include the zero polynomial, which has all
its coefficients equal to zero. There are natural rules of addi-
tion and scalar multiplication in P n (R), namely the familiar
ones of elementary algebra: to add two polynomials add cor-
responding coefficients; to multiply a polynomial by a scalar
c, multiply each coefficient by c. Using these operations, we
obtain the vector space of all real polynomials of degree less
than n.
This example could be varied by allowing polynomial of
arbitrary degree, thus yielding the vector space of all real poly-
nomials
P(R).

As usual R may be replaced by any field of scalars here.
Common features of vector spaces
The time has come to identify the common features in
the above examples: they are:
(i) a non-empty set of objects called vectors, including a
"zero" vector;
(ii) a way of adding two vectors to give another vector;
(iii) a way of multiplying a vector by a scalar to give a
vector;
(iv) a reasonable list of rules that the operations
mentioned(ii) and (iii) are required to satisfy.

We are being deliberately vague in (iv), but the rules should
correspond to properties of matrices that are known to hold
in R n andM m > n (R).