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 $(a+b) + (a+b+c)x$.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to join this community
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 $a(1+x) + (b-a)x$.
Also, as a side note, it would be better if you used mathjax to type the content from the image, or at least properly cropped the image.