# Affine hull of two points in R4 [closed]

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 .

## closed as off-topic by Lord_Farin, user91500, Najib Idrissi, MathOverview, yoknapatawphaOct 10 '15 at 18:06

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Lord_Farin, user91500, MathOverview, yoknapatawpha
If this question can be reworded to fit the rules in the help center, please edit the question.

• It is the straight line through these points. As you're in dimension $4$ you need $3$ linearly independent equations. – Bernard Oct 10 '15 at 10:35
• i conclude in this form which describe the above affine hull : c1[0;-1;0;1]+ [1;4;2;3] what can i do to make a linear equation from this ? – KostasC Oct 10 '15 at 16:46

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 $2$.