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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Construct a dual basis that extracts nonstandard coordinates.
- Justify the conclusion "φ₁(4,2)=3 and φ₂(4,2)=1" using the stated assumptions.
Before you start
Bases and linear functionals.
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 b₁=(1,1), b₂=(1,−1), find functionals φ₁,φ₂ with φᵢ(bⱼ)=δᵢⱼ.
Why this math matters
Construct a dual basis that extracts nonstandard coordinates. 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 vector space is R².
- Functionals are real-valued and linear.
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.
Identify linear measurements as dual vectors
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For b₁=(1,1), b₂=(1,−1), find functionals φ₁,φ₂ with φᵢ(bⱼ)=δᵢⱼ.
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
Let φ₁(x,y)=ax+by; a+b=1 and a−b=0
The first dual functional must select the first basis coordinate.
Work through the mathematics
φ₁=(x+y)/2; φ₂=(x−y)/2
Solving the analogous equations gives the second coordinate extractor.
Check the conclusion
φ₁(4,2)=3 and φ₂(4,2)=1
Dual functionals recover coefficients without changing the underlying vector.
The result
φ₁(4,2)=3 and φ₂(4,2)=1
Dual functionals recover coefficients without changing the underlying vector.
Common mistakes to catch
- A dual vector is a linear map, not simply a copied coordinate arrow.
- Dual bases depend on the chosen primal basis.
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 the dual of the standard basis?
Show a hint
Extract each usual component.
Reveal answer and explanation
The coordinate maps (x,y)↦x and (x,y)↦y
Each map equals one on its own basis vector and zero on the other.
Practice 2
Is v↦||v|| linear?
Show a hint
Test negation.
Reveal answer and explanation
No
A norm has ||−v||=||v||, unlike a nonzero linear functional.
Take the idea with you
Represent a calibrated sensor reading as a linear functional on a state vector.
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: Build a bilinear interaction from elementary tensors
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