Affine hull of two points in R4

Question

I try to describe an affine hull of two points (1,3,2,4) and (1,4,2,3) so i try to make the linear equation which describe it .

Answer 1

You need $3$ linear equations.

Let $A=(1,3,2,4)$, $B=(1,4,2,3)$, $M=(x,y,z,t)$.

One way to go is to write the vectors $\overrightarrow{AM}= (x-1,y-3,z-2,t-4)$ and $\overrightarrow{AB}=(0,1,0,-1)$ are colinear, i.e. $$\begin{cases}x-1=0\\ \dfrac{y-3}1=\dfrac{t-4}{-1}\\z-2=0\end{cases}\quad\text{whence}\quad\begin{cases}x=1\\z=2\\y+t=7\end{cases}$$

You also can apply row reduction to the matrix: $$\begin{bmatrix}x-1&0\\y-3&1\\z-2&0\\t-4&-1\end{bmatrix}$$ to obtain the compatibility conditions for the matrix to have rank $1$. This amounts to saying the $2\times2$ minors of the matrix are equal to $0$.

Last method, if you know projective space: write the matrix: $$\begin{bmatrix}x&1&1\\y&3&4\\z&2&2\\t&4&3\\1&1&1\end{bmatrix}$$ (i.e. add a 5th coordinate equal to $1$) and write the conditions for the matrix to have rank $2$.