Finite Dimensional Vector Spaces

Consider the problem of finding the set of points of intersection of the two planes $2 x + 3 y + z + u=0$ and $3x + y + 2z + u= 0.$
Let $V$ be the set of points of intersection of the two planes. Then $V$ has the following properties:

1. The point $(0,0,0,0)$ is an element of $V.$
2. For the points $(-1,0,1,1)$ and $(-5,1,7,0)$ which belong to $V;$ the point $(-6, 1, 8,1) = (-1,0,1,1)+(-5,1,7,0) \in V.$
3. Let ${\alpha}\in {\mathbb{R}}.$ Then the point ${\alpha}(-1,0,1,1) = (-{\alpha}, 0, {\alpha}, {\alpha})$ also belongs to $V.$

Similarly, for an $m \times n$ real matrix $A,$ consider the set $V,$ of solutions of the homogeneous linear system $A {\mathbf x}= {\mathbf 0}.$ This set satisfies the following properties:

1. If $A {\mathbf x}= {\mathbf 0}$ and $A {\mathbf y}= {\mathbf 0},$ then ${\mathbf x}, {\mathbf y}\in V.$ Then ${\mathbf x}+ {\mathbf y}\in V$ as $A ({\mathbf x}+ {\mathbf y}) = A {\mathbf x}+ A {\mathbf y}= {\mathbf 0}+ {\mathbf 0}= {\mathbf 0}.$ Also, ${\mathbf x}+ {\mathbf y}= {\mathbf y}+ {\mathbf x}.$
2. It is clear that if ${\mathbf x}, {\mathbf y}, {\mathbf z}\in V$ then $({\mathbf x}+{\mathbf y})+{\mathbf z}= {\mathbf x}+({\mathbf y}+{\mathbf z}).$
3. The vector ${\mathbf 0}\in V$ as $A {\mathbf 0}= {\mathbf 0}.$
4. If $A {\mathbf x}= {\mathbf 0}$ then $A (-{\mathbf x}) = - A {\mathbf x}= {\mathbf 0}.$ Hence, $-{\mathbf x}\in V.$
5. Let ${\alpha}\in {\mathbb{R}}$ and ${\mathbf x}\in V.$ Then ${\alpha}{\mathbf x}\in V$ as $A ({\alpha}{\mathbf x}) = {\alpha}A {\mathbf x}= {\mathbf 0}.$

Thus we are lead to the following.