Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Derive normal equations from an orthogonal residual condition.
- Justify the conclusion "Aᵀ(Ac−b)=0 and F″=6>0" using the stated assumptions.
Before you start
Matrices, derivatives, and sums of squares.
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
Fit one constant c to measurements 2,5,8 by minimizing squared residuals.
Why this math matters
Derive normal equations from an orthogonal residual condition. 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:
- All three observations have equal weight.
- The fitted model is constrained to be constant.
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.
Fit a constant when measurements disagree
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Fit one constant c to measurements 2,5,8 by minimizing squared residuals.
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
F(c)=(c−2)²+(c−5)²+(c−8)²
The objective gives large residuals greater influence.
Work through the mathematics
F′(c)=6c−30=0 ⇒ c=5
The stationary point balances the signed residuals.
Check the conclusion
Aᵀ(Ac−b)=0 and F″=6>0
With A=(1,1,1)ᵀ, the normal equation gives the unique minimum, whose residual sum is zero.
The result
Aᵀ(Ac−b)=0 and F″=6>0
With A=(1,1,1)ᵀ, the normal equation gives the unique minimum, whose residual sum is zero.
Common mistakes to catch
- Minimizing squared error is a modeling choice.
- A zero residual sum does not mean every residual is zero.
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
Fit a constant to 1,1,7,7.
Show a hint
Set the residual sum to zero.
Reveal answer and explanation
c=4
The equation 4c−16=0 gives the mean.
Practice 2
Would absolute-error fitting always give this same constant?
Show a hint
Compare mean and median.
Reveal answer and explanation
No
Absolute-error minimizers are medians; squared-error minimizers are means.
Take the idea with you
Compare squared-error and absolute-error summaries for data containing an outlier.
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: Find directions a coupled system preserves
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