Say I have a Line $P$ that cuts across 2 points $(0, 1, -1)$ and $(1,-1,0)$ in a space of $\mathbb{R}^3$.
If those were 2 vectors, I could say they span a plane, 3 vectors then the entire space. Since they are points and not vectors, what does a line across these points in $\mathbb{R}^3$ mean? I cannot say that it spans a line in the row/column space because this is a line.
I tried to find out the equations to this line by doing this: $\begin{bmatrix} 0 & 1 & -1 & 1\\ 1 & -1 & 0 & 1 \end{bmatrix}\begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix}=\begin{bmatrix} 0\\ 0 \end{bmatrix}$ $\begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix}= t\begin{bmatrix} 2\\ 1\\ 0\\ -1 \end{bmatrix} + k\begin{bmatrix} -1\\ 0\\ 1\\ 1 \end{bmatrix}, \; \; k,t \in \mathbb{R}$ Then I get 2 equations from the above: $2x+y=1$ $-x+z=-1$
Do these 2 equations represent the line $P$ that cut across the points $(0, 1, -1)$ and $(1,-1,0)$? But how do they represent because each of the 2 equations themselves is a line on their own.
I could continue to derive from the 2 equations to find out the solution set to $x$, $y$ and $z$: $\begin{bmatrix} 2 & 1 & 0\\ -1 &0 & 1 \end{bmatrix} \begin{bmatrix} x\\ y\\ z \end{bmatrix} = \begin{bmatrix} 1\\ -1 \end{bmatrix}$ $\begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix} + c\begin{bmatrix} 1\\ -2\\ 1 \end{bmatrix}$
Again, at this point, I am also very confuse what this set of solution is representing. I know that this set of solution is for the 2 equations above but does it mean that if in the column space, by moving in any amount of $ $$\begin{bmatrix} x\\ y\\ z \end{bmatrix} =\begin{bmatrix} 1\\ -1\\ 0 \end{bmatrix} + c\begin{bmatrix} 1\\ -2\\ 1 \end{bmatrix}$ would let me reach the line $P$?