Let $ \ \large \mathbb{R}[x]_{<4} \ $ denote the space of polynomials in $ \ \mathbb{R} \ $

Question

Let $ \ \Bbb R[x]_{<4} \ $ denote the space of polynomials in $ \ \Bbb R \ $ of degree less or equal to $ \ 3 \ $.

Can we find a basis $ \ \{p_1, \ p_2, \ p_3 , \ p_4 \} \ $ of $ \ \Bbb R[x]_{<4} \ $ such that none of $ \ p_i \ $ has degree $ \ 1 \ $.

Answer:

Since $ \ \Bbb R[x]_{<4} \ $ denote the space of polynomials in $ \ \Bbb R \ $ of degree less or equal to $ \ 3 \ $ , the standard basis is given by

$ \mathcal B=\{1,x,x^2,x^3 \} \ $

But how can I replace $ \ x \ $ in $ \ \mathcal B \ $ with another independent vector in order to get the required basis because $ \ p(x)=x \ $ has degree $ \ 1 \ $.

Help me out of this

Answer 1

Sure. For instance, just add $x^3$ to each element of the standard basis: $$\{1+x^3,x+x^3,x^2+x^3,2x^3\}.$$

Answer 2

You may consider the following basis

$$\{1, x+x^2, x^2, x^3\}$$

Any element has degree different than one. Of course you may obtain $x$ as $(x+x^2)-x^2$.