We are asked to find the standard matrix $A$ for $T$:
Consider the transformation $T : \mathbb{R^3} \rightarrow \mathbb{R^4}$ given by
$$T(x_1, x_2, x_3) = (x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3)$$ for every $$(x_1,x_2,x_3) \in \mathbb{R^3}$$
I am confused. I do not know how to start the problem. I have researched the description but I cannot find anything definitive for this type of problem.
Update:
$$ A = \left[\begin{array}{rrr} 1 & 1 & 1\\ 0 & 1 & 1\\ 3 & 1 & 0\\ 0 & 2 & 1\\ \end{array}\right] $$
$$ ref(A) = \left[\begin{array}{rrr} 1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1\\ 0 & 0 & 0\\ \end{array}\right] $$
From $T(e_1)=(1,0,3,0)^T, T(e_2)=(1,1,1,2)^T, T(e_3)=(1,1,0,1)^T$
Is this correct? Or am I completely lost?
How would I find the dimension of the kernel of $T$ and the range of $T$?