<s>[INST] Matrix linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^4$. Find the matrix representation Say that $T : \mathbb{R}^3 \to \mathbb{R}^4$ is linear and satisﬁes $T(1,1,1) = (3,2,0,1)$, $T(1,1,0) = (2,1,3, −1)$ and $T(1,0,0) =(5, −2,1,0)$. Find $T(x, y, z)$ and then ﬁnd the matrix representation for $T$.
I am not sure how to proceed. I thought of two approaches, one the matrix $A$ from $Ax=b$ could be expressed as
$$A=\begin{pmatrix}
3 & 2 & 5\\
2 & 1 & -2\\
0 & 3 & 1\\
1 & -1 & 0
\end{pmatrix}$$
Then possibly solve for $Ax=0$?
Or since $T(1,0,0)=v_3$ etc, we can also write $T(x,y,z)=x\cdot v_1+y \cdot v_2+z \cdot v_3$, but I am not sure how to find $v_1$, $v_2$ or $v_3$. 
Appreciate your help. 
 [/INST] $T(0,0,1) = T(1,1,1) - T(1,1,0) = (1, 1, -3, 2)$, and $T(0,1,0) = T(1,1,0) - T(1,0,0) = (-3, 3, 2, -1)$, and $T(1,0,0) = (5, -2, 1, 0)$. So 
\begin{align*}T(x,y,z) &= xT(1,0,0) + yT(0,1,0) + zT(0,0,1) \\
&= x(5,-2,1,0) + y(-3,3,2,-1) + z(1,1,-3,2) \\
&= (5x-3y+z, -2x+3y+z, x+2y-3z, -y+2z).\end{align*} And from this the matrix $A$ is:
\begin{pmatrix}
5  & -3 & 1 \\
-2 & 3 & 1 \\
1 & 2 & -3 \\
0 & -1 & 2 \\
\end{pmatrix}
You can continue.
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