I have never studied linear algebra at undergrad, now I need to take a test that has questions related to it. I'm struggling to solve a seemingly simple question:
if perhaps this has already been solved, direct me to the right link. Otherwise, my attempt was to combine the three matrices being transformed into one 3x3 matrix and then finding the RREF for that matrix, which does not sit well with me as correct.
Any help will be appreciated.
First show that $$\{(-1,2,0), (-1,-1,1), (1,0,1)\}$$ is a basis for $\mathbb{R}^3$
Then find $a_1,a_2,a_3 \in \mathbb{R}$ such that $$a_1 (-1,2,0) + a_2(-1,-1,1) + a_3(1,0,1) = e_1$$ Thus, $$a_1 g(-1,2,0)+a_2g(-1,-1,1) + a_3 g(1,0,1) = g(e_1) = (-1, 3, 0)$$ because $g$ is linear operator (you already know who are $g(-1,2,0), g(-1,-1,1),g(1,0,1)$). ;)
Repeat for $b_1, b_2, b_3, c_1, c_2, c_3 \in \mathbb{R}$ such that $$b_1 (-1,2,0) + b_2(-1,-1,1) + b_3(1,0,1) = e_2 \longrightarrow g(e_2) = (0,1,2)$$ $$c_1 (-1,2,0) + c_2(-1,-1,1) + c_3(1,0,1) = e_3 \longrightarrow g(e_2) = (2,2,1)$$