Linear Algebra Geometric Planes

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SUMMARY

The discussion focuses on the geometric interpretation of a homogeneous system of three equations in three unknowns, represented by three planes in space that intersect at the origin. The relationship between the solution set and the number of free variables in the Reduced Row Echelon Form (RREF) of the coefficient matrix A is established. Specifically, if the RREF has three pivot columns, the only solution is the origin; if there is one free variable, the solution set forms a line through the origin; and if there are two free variables, the solution set forms a plane through the origin.

PREREQUISITES
  • Understanding of homogeneous systems of equations
  • Familiarity with Reduced Row Echelon Form (RREF)
  • Basic knowledge of geometric interpretations of linear algebra
  • Ability to visualize intersections of planes in three-dimensional space
NEXT STEPS
  • Study the properties of homogeneous systems of equations
  • Learn about the geometric interpretation of RREF in linear algebra
  • Explore the concept of free variables and their impact on solution sets
  • Practice visualizing intersections of planes using graphing tools
USEFUL FOR

Students of linear algebra, educators teaching geometric interpretations of systems of equations, and anyone seeking to deepen their understanding of RREF and its applications in geometry.

lina29
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For a homogeneous system of 3 equations in 3 unknowns (geometrically
this is 3 planes in space all containing the origin) describe the relationship between
the (three) geometric possibilities for the solution set and the number of free variables (non
pivots) in RREF(A) where A is the coefficient matrix.

I understand what RREF stands for. However, I don't understand how to approach the question or what it is asking
 
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Obviously, three planes all passing through the origin have the origin in common. It is possible that the origin is the only point in common. But it is also true that there is a whole line through the origin lying in all three plane, or an entire plane in all three planes. Try drawing pictures to see what the geometric relationships would be and what effect they would have on the RREF matrix.
 

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