Generalized Linear equation of a cube

[Leo Authersh]

As per my understanding, a linear equation with two variables form a line segment (ax=by+c or ax+by=c) and linear equation with three variables form a plane (ax=by+cz+d or ax+by+cz=d). Am I right? And if I am right, does an equation with four variables form a cube?

 

[Orodruin]

As per my understanding, a linear equation with two variables form a line segment (ax=by+c or ax+by=c) and linear equation with three variables form a plane (ax=by+cz+d or ax+by+cz=d). Am I right?

A single linear equation always describes a space of one dimension lower thanthe one you started with. If you have three variables, it is a two-dimensional plane and if you have two it is a one-dimensional line. If you have four it is a three-dinensional hyperplane and so on.

Note that ax+by=d might be an equation in three variables - the coefficient of z could just happen to be zero. You need context to know this.

And if I am right, does an equation with four variables form a cube?

No.

 

[FactChecker]

@Leo Authersh , if by "cube" you really mean three-dimensional hyperplane in 4-dimensional space, then that is right. The term "cube" is wrong. It implies a region bounded on all sides, but the hyperplane is unbounded.

 

[WWGD]

No; translate your equation $ax_1+bx_2+cx_3+dx_4 =e$ so that it goes through the origin. As a linear object, it is closed under addition, linear combination. Find the mid point of two ( say, to simplify, opposite, meaning non-adjacent; not sharing an edge) faces of a cube. The midpoint will not lie on the cube. The cube is not a subspace, unlike the set of points described above.

 

[Leo Authersh]

@Leo Authersh , if by "cube" you really mean three-dimensional hyperplane in 4-dimensional space, then that is right. The term "cube" is wrong. It implies a region bounded on all sides, but the hyperplane is unbounded.

Thank you. Is it possible to visualize a 3 dimensional hyperplane?

 

[FactChecker]

Thank you. Is it possible to visualize a 3 dimensional hyperplane?

It's just like 3-dimensional space. The harder part is picturing the 4-dimensional space (of the 4 variables) that it is in.

Consider the 4-dimensional space-time (x,y,z,t). Suppose you have one simple equation like t=5. Then the hyperplane it defines is the simple (x,y,z,5) set of 3-dimensional space at time t=5. Now consider a more complicated equation like x+y+z+t = 5. All that does is make a "tilted" hyperplane of points satisfying that equation. (Just like in two dimensions x+y=5 makes a sloped 1-dimensional sloped line in (x,y) )