Generalized Linear equation of a cube

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Discussion Overview

The discussion revolves around the nature of linear equations in multiple variables, specifically exploring whether an equation with four variables can be interpreted as forming a cube. Participants examine the dimensionality of linear equations and the implications of terminology used in describing geometric objects in higher dimensions.

Discussion Character

  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that a linear equation with two variables describes a line segment, while one with three variables describes a plane.
  • One participant proposes that an equation with four variables might form a cube, but others challenge this notion.
  • It is noted that a single linear equation describes a space of one dimension lower than the number of variables, suggesting that four variables correspond to a three-dimensional hyperplane.
  • Another participant clarifies that the term "cube" is misleading, as it implies a bounded region, whereas a hyperplane is unbounded.
  • Participants discuss the visualization of a three-dimensional hyperplane within four-dimensional space, using examples like equations in space-time to illustrate the concept.

Areas of Agreement / Disagreement

There is disagreement regarding the interpretation of a four-variable linear equation as a cube. While some participants clarify that it represents a hyperplane, others initially suggest the possibility of it being a cube. The discussion remains unresolved on the terminology and implications of these concepts.

Contextual Notes

Participants highlight the importance of context in understanding the dimensionality of equations and the definitions of geometric terms. There is an acknowledgment that certain equations can be interpreted differently based on their coefficients and the dimensional space they occupy.

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?
 
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Leo Authersh said:
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.

Leo Authersh said:
And if I am right, does an equation with four variables form a cube?
No.
 
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@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.
 
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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.
 
FactChecker said:
@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?
 
Leo Authersh said:
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) )
 
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