How to visualize a three dimensional hyperplane?

Question

In one of the exercise of Linear Algebra it is mentioned that the columns of following matrix lie in a three dimensional hyperplane $$ M= \begin{bmatrix} 1 & 0 & 0 & -1\\ -1 & 1 & 0 & 0\\ 0 & -1 & 1 & 0\\ 0 & 0 & -1 & 1 \end{bmatrix} \ $$ I know that the three dimensional hyperplane is defined as $$a_1x_1+a_2x_2+a_3x_3+a_4x_4=b$$ But how can we prove that the columns of above matrix $M$ lie in a three dimensional hyperplane? And what should be the values of $a_1,a_2,a_3,a_4,b$ for the hyperplane? How to visualize this hyperplane? Any help in this regard will be much appreciated. Thanks in advance.

Answer 1

We have that $$c_4=-(c_1+c_2+c_3)$$ with $c_1$,$c_2$ and $c_3$ linearly independent, thus the columns of $M$ span a subspace of dimension $3$ that is an hyperplane in $\mathbb{R^4}$.

Answer 2

Obviously the first three column vectors are linearly independent and the fourth vector is the sum of those three, therefore their span is $3$-dimensional. So $(1,1,1,1)$ is a normal vector for the hyperplane, hence $a_1=a_2=a_3=a_4=1$ and $b=0$.