Describe $(1,1,1,3)+lin \left\{ (4,-2,3,-2),(-2,0,0,1) \right\}$ with the system of equations I did this task and I need to check if I'm doing the right thing.My try:Let $((1,1,1,3)+lin \left\{ (4,-2,3,-2),(-2,0,0,1) \right\})=H$Then $T(H)$: $$\begin{bmatrix}4 & -2 & 3 & -2 & | & 0 \\-2 & 0 & 0 & 1 & | & 0 \end{bmatrix} \rightarrow \begin{bmatrix}2 & 0 & 0 & -1 & | & 0 \\0 & -2 & 3 & 0 & | & 0 \end{bmatrix}$$ So I have: $$ \begin{cases} 2x_{1}-x_{4}=0 \\ -2x_{2}+3x_{3}=0 \end{cases} $$
$$\begin{cases} x_{4}=2x_{1} \\ x_{3}=\frac{2}{3} x_{2} \end{cases} $$That is why: $$T(H)=lin \left\{(1,0,0,2), (0,3,2,0)\right\}$$Hence:$$H:  \begin{cases} y_{1}+2y_{4}=b \\ 3y_{2}+2y_{3}=c \end{cases} $$ $$b=1 + 2 \cdot 3 =7$$ $$c=3 \cdot 1+ 2 \cdot 1=5$$ So my sollution:$$H:  \begin{cases} y_{1}+2y_{4}=7 \\ 3y_{2}+2y_{3}=5 \end{cases} $$

Is this the correct result?
 A: Thr equations for T(H) that you should obtain when you abandon matrix notation are 
$$\left\{\begin{array}{l} 2x_1 - x_4 = 0 \\ -2x_2+3x_3 =0 \end{array}\right. $$
Can you continue from this point?
A: First of all, you can’t combine the equations as you’ve done at the end. The affine space that you’re trying to describe is two-dimensional, but the solution set of a single linear equation in four variables is three-dimensional. You need a system of at least two implicit linear equations to describe the set properly.  
You’ve also made a computational error somewhere along the way: your system of equations is not satisfied by all of the points in the given set. If you substitute $(1,1,1,3)+\lambda(4,-2,3,-1)+\mu(-2,0,0,1)$ into the left-hand sides of the two equations, you get $23+6\lambda$ and $1-2\lambda$, respectively, instead of the required results. Observe that you’ve made an error right off the bat: the second row of your reduced matrix corresponds to the equation $-2x_2+3x_3=0$, i.e., you’ve mistakenly written $x_1$ instead of $x_2$. That error cascades through the rest of you calculations.  

Your updated solution looks correct to me, but you can check it for yourself: does $(1,1,1,3)+\lambda(4,-2,3,-1)+\mu(-2,0,0,1)$ satisfy both equations?
