Can someone help ? I have this answer of linear transformation.

Question

Onto? What we have in class that if $n=2$ and $m=3$ that clear $2<3$ ,$T$ will be not onto. He said make three point u have with three variable $=(y,z)$ .then u will have one free variable that will be expressed by $x_1,x_2,x_3$... I do it $x_1$ will free variable but I get very complicated answer,and does the question mean what the condition to be onto or 1-1? last answer i have i do not make sure can some one help

Answer 1

If $b_1,b_2\in\Bbb R$, the equations $$x_1+2x_2+3x_3=b_1\ ,\quad -x_1-4x_3=b_2$$ always have a solution. Therefore $T$ is onto.