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Answer:
(a) Three polynomials of degree 1 with real coefficients belong to the set , then:
(b) Three polynomials of degree 1 with real coefficients that hold the relation belong to the set
. The relation between the coefficients is equivalent to
, then:
Step-by-step explanation:
(a) Three polynomials of degree 1 with real coefficients belong to the set , then:
A vector space is any set whose elements hold the following axioms for any and
and for any scalar
and
:
- There is the <em>zero element </em>such that:
- For all element
of the set, there is an element
such that:
Let's proof each of them for the first set. For the proof, I will define the polynomials ,
and
and the scalar
and
.
and defining
and
, we obtain
which is another polynomial that belongs to
- A null polynomial is define as the one with all it coefficient being 0, therefore:
- Defining the inverse element in the addition as
, then
With this, we proof the set is a vector space with the usual polynomial addition and scalar multiplication operations.
(b) Three polynomials of degree 1 with real coefficients that hold the relation belong to the set
. The relation between the coefficients is equivalent to
, then:
Let's proof each of axioms for this set. For the proof, I will define again the polynomials ,
and
and the scalar
and
. Again the relation
between the coefficients holds
and considering the coefficient relation and defining
which is another element of the set since it is a degree one polynomial whose coefficient follow the given relation.
The proof of the other axioms can be done using the same logic as in (a) and checking that the relation between the coefficients is always the same.