# Range of linear mapping [closed]

In this example, why is the range given as $$Range(L)= (1+x, x)$$ and not as simply $$(1, x)$$.

I thought that it would be $$(1,x)$$ since we can use a linear combination of $$1$$ and $$x$$ to express the form $$(a+b) + (a+b+c)x$$.

Is $$P_1$$ the set of polynomials of degree $$\leq 1$$?
In any case, the span of $$(1,x)$$ and $$(1+x,x)$$ is the same. That is to say, any polynomial that you can write as a linear combination of $$1$$ and $$x$$ you can also write as a linear combination of $$1+x$$ and $$x$$ and vice versa.
For example, we can write the polynomial $$a+bx$$ as $$a(1+x) + (b-a)x$$.