If I have a basis of:
$(1, x+1, (x+1)x)$
how do I write the basis in terms of the standard basis $(1, x, x^2)$?
The answers tell me I'm supposed to get 3 vectors:
$[1,0,0] , [1,1,0], [0,1,1]$
But I'm not sure how to arrive to that point, any help is appreciated!
You should express each basis vector as a linear combination of $(1,x,x^2)$ and take the corresponding coefficients. For example,
$$ (x+1)x = x^2 + x = 0 \cdot 1 + 1 \cdot x + 1 \cdot x^2 $$
so the corresponding vector representing $(x+1)x$ with respect to the basis $(1,x,x^2)$ is (in your notation) $[0,1,1]$.