## Coordinates in the space of functions

In the space of real polynomials of degree at most four with the basis $$X=(x^4+x^3,\ x^3+x^2,\ x^2+x,\ x+1,\ x^4+1)$$ determine coordinates $$[f]_X$$ of the following vectors $$f$$:

• #### Variant 1

$$f(x)=x^4-1$$.

• #### Variant 2

$$f(x)=x^4+x^3+x^2+x+1$$.

• #### Variant 3

$$f(x)=x^4+x^2+1$$.

• #### Variant 4

$$f(x)=x^3+x$$.