Solving Ax=0 | Summary and Q&A

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July 25, 2018
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Solving Ax=0

TL;DR

This video explains how to solve homogeneous linear systems and introduces the concept of non-homogeneous linear systems.

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Key Insights

  • ☺️ Solving a homogeneous linear system involves finding solutions where all variables satisfy the equation x - 5y + 2z = 0.
  • πŸ«₯ Two planes in R^3 can either intersect along a line or be parallel.
  • πŸ˜₯ The specific form of all points in the plane S is obtained by adding the direction vector to any point in the plane S_0.
  • πŸ₯Ά Homogeneous linear systems can be solved by parameterizing the free variables and finding particular solutions.
  • 😫 The set of all solutions to a homogeneous linear system forms a vector space.
  • ✈️ The concept of parallel planes and direction vectors is important in understanding the relationship between homogeneous and non-homogeneous linear systems.
  • 🚱 Solving the homogeneous linear system is a necessary step in solving non-homogeneous linear systems.

Transcript

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Questions & Answers

Q: What is the difference between a homogeneous and non-homogeneous linear system?

A homogeneous linear system has a constant term of 0 in all equations, while a non-homogeneous linear system has non-zero constant terms.

Q: How can we determine the position of two planes in R^3?

Two planes in R^3 can either intersect along a line or be parallel.

Q: How can we find the line of intersection between two planes?

By setting the equations of both planes equal to each other and solving the resulting system of equations.

Q: How do we express all points in the plane S?

All points in the plane S can be expressed as the sum of a specific point P_0 and a linear combination of the direction vector connecting S_0 to S.

Summary & Key Takeaways

  • The problem involves solving a homogeneous linear system and understanding its relation to a non-homogeneous linear system.

  • The set S consists of all points that satisfy the equation x - 5y + 2z = 9, while S_0 consists of points that satisfy x - 5y + 2z = 0.

  • Two planes can either intersect along a line or be parallel, and in this case, S and S_0 are parallel planes.

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