Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Measure how unequal matrix scales amplify input errors.
- Justify the conclusion "κ₂(A)=1000" using the stated assumptions.
Before you start
Matrix inversion and Euclidean norms.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
For A=diag(1,0.001), compare b=(1,0) with b̃=(1,0.001).
Why this math matters
Measure how unequal matrix scales amplify input errors. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The second diagonal entry is nonzero.
- Conditioning concerns the problem, not only the solver.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Separate a small residual from a trustworthy solution
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For A=diag(1,0.001), compare b=(1,0) with b̃=(1,0.001).
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
A⁻¹=diag(1,1000); x=(1,0)
The second coordinate is magnified by a factor of one thousand.
Work through the mathematics
x̃=A⁻¹b̃=(1,1)
An input perturbation of size 0.001 changes the solution by size one.
Check the conclusion
κ₂(A)=1000
The condition number warns about worst-direction sensitivity even though the arithmetic example is exact.
The result
κ₂(A)=1000
The condition number warns about worst-direction sensitivity even though the arithmetic example is exact.
Common mistakes to catch
- Do not confuse numerical stability with conditioning.
- The worst-case bound need not be attained by every perturbation.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is κ₂(diag(2,2))?
Show a hint
Divide largest by smallest singular value.
Reveal answer and explanation
1
Equal stretching in every direction avoids directional amplification of relative errors.
Practice 2
Does a tiny residual guarantee a tiny solution error?
Show a hint
Error satisfies A(x̃−x)=residual.
Reveal answer and explanation
Not for an ill-conditioned matrix
Applying a large inverse can magnify the residual.
Take the idea with you
Identify which direction of measurement noise would be most damaging in this model.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Track composition order in permutations
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