Question

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

Answer 1

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$4\times 4$

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.