Polynomials as Vectors

See Polynomials

Continuing with what was discussed in Vector Spaces

Polynomial as a Vector

Definition

For all non-negative integers , the set:

denotes the set of all real polynomials of degree at most .

(see set builder notation)

Remark

We express the degree as :
denotes constant polynomials, denotes both constant AND first degree polynomials, and so on.

This means that
(see subset).

Theorem

With the operations of addition and scalar multiplication, is a vector space for each non-negative integer .

Zero Polynomial

Definition

We denote the zero polynomial by . (see zero polynomial).

Equality, Addition, and Scalar Multiplication

Polynomial Equality

Definition

The polynomials and in are equal if and only if for all

Addition and Multiplication of Polynomials

Definition

Let and

Then:

where for and .
That is, when adding polynomials, we add their coefficients.

where
for and where for , and for
That is, when multiplying polynomials, we have to multiply each term of one polynomial by each term of the other polynomial, INCLUDING the indeterminants (and their powers add up).

This is also known as expanding.

(See Summation and Product Notation)

Remark

This is a super technical definition which is practically useless. Adding and multiplying polynomials is very intuitive.

Example

In , for and :

In , for and

In , for and (see congruence class)

Standard Basis for Polynomials

(see standard basis)

Definition

The set is called the standard basis for

Dimension of Polynomial

(see dimension)

Definition