Show that x^3 +x, x^2- x, x +1, x3 + 1 form a basis for p3

Answer  with Step-by-step explanation:

We are given that

Therefore, the dimension of is 4.

We have to prove that form basis for

We will  prove basis of polynomial by the help of matrix

We make a matrix coefficient of

When we prove basis for

It means every element of 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

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

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 .

Therefore, all vectors are linearly independent .Hence, they span 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 .Hence,  every element is of is a linear combination of given linearly independent vectors.

Hence, proved.

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step-by-step explanation:

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step-by-step explanation: