Question

Determine if the set of polynomials {x^2 –2x+1, 2x² + 3x -4,-x2+x+5) is a linearly independent set in P2. Is it a basis for P? Why or why not?

Answer 1

Answer with Step-by-step explanation:

We are given that a set of polynomials ${x^2-2x+1,2x^2+3x-4,-x^2+x+5}$

We have to find that given set is a linearly independent set in$P_2$

and given set is a basis for $P_2$ or not.

Matrix of given set of polynomials

A=$\left[\begin{array}{ccc}1&-2&1\\2&3&-4\\-1&1&5\end{array}\right]$

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

$P_2=x^2$

Element of $P_2$ is of the form

$ax^2+bx+c$

Every element of $p_2$ is a linear combination of given set of polynomials.

Hence, given set is linearly independent in $p_2$ .

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 $P_2$.

Therefore, given set is  as basis for $p_2$ because the set is linearly independent and spanned $P_2$.