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Show that x^3 +x, x^2- x, x +1, x3 + 1 form a basis for P3

User Alexnnd
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Answer with Step-by-step explanation:

We are given that


x^3+x,x^2-x.x+1,x^3+1


P_3=x^3

Therefore, the dimension of
P_3 is 4.

We have to prove that
x^3+x,x^2-x,x+1,x^3+1 form basis for
P_3

We will prove basis of polynomial by the help of matrix

We make a matrix coefficient of
x^3,x^2,x, costant\; value


\left[\begin{array}{cccc}1&0&1&0\\0&1&-1&0\\0&0&1&1\\1&0&0&1\end{array}\right]

When we prove basis for
P_3

It means every element of
P_3 is a linear combination of basis.

We prove basis then we should prove given vectors are linearly independent .If given vectors are linearly independent then they form basis for
P_3

We find rank

Rank= Number of non zero rows and no row is a linear a combination of other rows.

In above matrix of order
4* 4

Any row or column is not a linear combination of other any two or more rows or columns .Therefore, the rank of matrix

Rank=4=Dimension of
P_3.

Therefore, all vectors are linearly independent .Hence, they span
P_3 because every linear independent set is a spanning set of a given vector space.

When they are linearly independent then they should form basis for
P_3 .Hence, every element is of
P_3 is a linear combination of given linearly independent vectors.

Hence, proved.

User Joel Teply
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