Understanding higher dimensions For every 2 different points in 3D space, there is exactly one straight line that goes between them. For every 3 points that are not on the same line in 3D space, there is exactly one surface that goes through all 3.
How can you express this mathematically? And how does this phenomenon translate to higher dimensions?
 A: We will start with 2 operations of vectors: $+,\cdot$:$$\vec{v}+\vec{u} =\begin{bmatrix}v_{1}\\
v_{2}\\
.\\
.\\
.\\
v_{n}
\end{bmatrix}+\begin{bmatrix}u_{1}\\
u_{2}\\
.\\
.\\
.\\
u_{n}
\end{bmatrix}=\begin{bmatrix}v_{1}+u_{1}\\
v_{2}+u_{2}\\
.\\
.\\
.\\
v_{n}+u_{n}
\end{bmatrix}$$
$$a\vec{v} =a\begin{bmatrix}v_{1}\\
v_{2}\\
.\\
.\\
.\\
v_{n}
\end{bmatrix}=\begin{bmatrix}av_{1}\\
av_{2}\\
.\\
.\\
.\\
av_{n}
\end{bmatrix}$$
where $v_{1,\cdots,n},u_{1,\cdots,n},a$ are, for simplicity's sake, real numbers.
We can say that $"nD"$ space is all the points that are linear combination of $n$ linearly independent vectors with $n$ elements, in other words: $\sum_{i=1}^{n}a_{i}e_{i}$ where $a_{i}\in\mathbb{R}$ and $e_{i}=\begin{bmatrix}
.\\
0\\
1\\
0\\
.
\end{bmatrix}$(has n elements and only the $i^{th}$ element is 1 while the others are 0)
This is what we call “the span of the standard basis”.
For example a point in $"3D"$ is:$$\begin{bmatrix}
x\\
y\\
z
\end{bmatrix}=xe_1+ye_2+ze_3$$
The vector in the sums I showed were the "axises", but it needs not to be so, if you have $n$ points that with $n-1$ of them you can't get to the last one using only the $2$ operations from above(linearly independent) you can replace the $e_i$ with your points.
Because the elements I take in those examples are all reals we call $"nD"$ as $\mathbb R^n$
So given $3$ points in a $"4D"$ that are linearly independent you can find the space those $3$ points create using the sum:
given $\vec v_1=\begin{bmatrix}
8\\4\\-3\\4
\end{bmatrix},\vec v_2=\begin{bmatrix}
-7\\-1\\0\\3
\end{bmatrix},\vec v_3=\begin{bmatrix}
83\\4\\7\\11
\end{bmatrix}$ the space that is created "inside" of the $"4D"$ is all the points that can be written as: $a_1\vec v_1+a_2\vec v_2+a_3\vec v_3,a_{1,2,3}\in\mathbb R$
