Answer with Step-by-step explanation:
We are given that a set of polynomials
We have to find that given set is a linearly independent set in
and given set is a basis for or not.
Matrix of given set of polynomials
A=
Linearly Independent set :If any row or any column is not a linear combination of other rows or columns then the set is linearly independent set.
Any row or column is not a linear combination of other rows or columns.Therefore, given set is a linearly independent set .
We know that
Element of is of the form
Every element of is a linear combination of given set of polynomials.
Hence, given set is linearly independent in .
If any set is basis for any vector space then it satisfied the following two conditions
1.Given set is linearly independent.
2.Every element of given vector space spanned by the given set.
Given set of polynomials are linearly independent and spanned every element of .
Therefore, given set is as basis for because the set is linearly independent and spanned
.