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Homogeneous Linear Systems
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
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Homogeneous Linear Systems
What is it?
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
In simple words
- Ax=0.
- Trivial solution
- Free variables if singular
- State the rule, show the working, and check the answer.
Important terms
HOMOGENEOUS LINEAR SYSTEMS: Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
HOMOGENEOUS: Ax=0.
What you should learn
- State the rule or meaning of homogeneous linear systems in easy words.
- Apply homogeneous linear systems in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Examples
- Trivial solution
- Free variables if singular
- Null space named
MAIN IDEA
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
EXAM USE
Write the rule first: Ax=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Recommended for Class 11
Full school-level explanation
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
Understand the idea, then apply it:
WHAT — HOMOGENEOUS LINEAR SYSTEMS
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
WHY — Why it matters
This rule is a separate learning unit inside linear systems: consistent or not. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Ax=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
MAIN IDEA
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
Remember
- always consistent
Examples
- Trivial solution
- Free variables if singular
- Null space named
EXAM USE
Write the rule first: Ax=0. Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Remember
- focused answer
- accurate support
Examples
- State the central idea
- Use one accurate detail
- Check the final response
How to answer a homogeneous linear systems question
How it works
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
Remember
- always consistent
- consistent = at least one x
- 0 = 1 means inconsistent
- Ax=0 always has 0
- always consistent
- focused answer
- accurate support
- always consistent
Deeper understanding
Go beyond memorising Homogeneous Linear Systems. Connect the definition to reasons, relationships, exceptions, and exam wording.
Common mistakes
Exam focus
Core Idea
What to Know
Example or Evidence
How to Answer
Exam focus
Quick revision
- always consistent
- consistent = at least one x
- 0 = 1 means inconsistent
- Ax=0 always has 0
- HOMOGENEOUS LINEAR SYSTEMS: Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Use
- HOMOGENEOUS: Ax=0.
Homogeneous is never inconsistent. Non-homogeneous can be.
For long answers on Homogeneous Linear Systems: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- always consistent
- consistent = at least one x
- 0 = 1 means inconsistent
- Ax=0 always has 0
Important formulas
- always consistent
- consistent = at least one x
- 0 = 1 means inconsistent
Common exam traps
- Homogeneous is never inconsistent. Non-homogeneous can be.
- Do not choose a formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Homogeneous is never inconsistent. Non-homogeneous can be.
- Mistake
Do not choose a formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes, restrictions, or skipped algebra steps.
- Mistake
Homogeneous is never inconsistent. Non-homogeneous can be.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
always consistent
consistent = at least one x
0 = 1 means inconsistent
Ax=0 always has 0
Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
consistent = at least one x
- Definition
- Ax=0. Always x=0. Non-trivial kernel if rank < n. Gaussian elimination on [A|0] never makes 0=1. Used for eigenvectors later (Grade 12 idea named only).
- Formula
- consistent = at least one x
- Example
- Trivial solution
- Critical warning
- Homogeneous is never inconsistent. Non-homogeneous can be.
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Exam Practice
Define
Q1. Define Homogeneous Linear Systems.
Explain
Q2. Explain: MAIN IDEA.
Example
Q3. Give one clear example of Homogeneous Linear Systems and say why it fits.
Mistake
Q4. State one common mistake students make about Homogeneous Linear Systems.
Exam note
Q5. Write a short exam note on Homogeneous Linear Systems: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.