Condition Number
Sensitivity of Ax = b to errors.
Key idea
Sensitivity of Ax = b to errors.
In depth
κ(A) = ‖A‖‖A⁻¹‖ measures how much errors in input are magnified in the solution. A large κ means the system is ill-conditioned, so small data changes produce big changes in the result.
Worked example
A nearly singular matrix like [1 1; 1 1.0001] has κ ≈ 40,000.
Formula
Real-world uses
- Warning when a computer solution may be inaccurate, such as in structural or survey calculations
Common mistake
Believing a tiny determinant alone proves poor conditioning; scale matters.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Condition Number?
Sensitivity of Ax = b to errors. It is a topic at the expert level in Linear Algebra.
What is the formula for Condition Number?
The key formula is κ(A) = ‖A‖‖A⁻¹‖.
Where is Condition Number used in real life?
Real-world examples: Warning when a computer solution may be inaccurate, such as in structural or survey calculations.
What should I learn before Condition Number?
Make sure you are comfortable with Norms of Vectors and Matrices, the previous topic in Linear Algebra.
What is a common mistake with Condition Number?
Believing a tiny determinant alone proves poor conditioning; scale matters.
How can I practise Condition Number?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Condition Number?
Move on to Power Iteration next.
Related expert topics in Linear Algebra
- Triple Product – Scalar triple product gives parallelepiped volume.
- Homogeneous Systems – Ax = 0 always has the trivial solution.
- Cofactor Expansion – Compute determinants recursively.
- Properties of Determinants – Multiplicative; row swaps change sign.
- Geometric Meaning of Determinants – Area or volume scaling factor.
- Cramer’s Rule for Systems – Solve systems using determinants.