Question

Suppose \({{\bf{p}}_{\bf{1}}}\), \({{\bf{p}}_{\bf{2}}}\), \({{\bf{p}}_{\bf{3}}}\), and \({{\bf{p}}_{\bf{4}}}\) are specific polynomials that span a two-dimensional subspace H of \({P_{\bf{5}}}\). Describe how one can find a basis for H by examining the four polynomials and making almost no computations.

Short answer

For the set \(\left\{ {{p_1},....,{p_4}} \right\}\), check for two polynomials that are not multiples of one another.

Step-by-step solution

Write the given information

The four polynomials are \({p_1}\), \({p_2}\), \({p_3}\), and \({p_4}\).

Check for the basis of H

For the set \(\left\{ {{p_1},....,{p_4}} \right\}\), check for two polynomials that are not multiples of one another.

Since these polynomials are linearly independent sets in a two-dimensional space, they form the basis of H by the basis theorem.