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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Expand an elementary tensor in a product basis.
- Justify the conclusion "The coefficient matrix is [[3,−1],[6,−2]], of rank one" using the stated assumptions.
Before you start
Vector spaces and bilinearity.
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
Expand (e₁+2e₂)⊗(3f₁−f₂) in the basis eᵢ⊗fⱼ.
Why this math matters
Expand an elementary tensor in a product basis. 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 two vector spaces each have the displayed two-element basis.
- Tensor products are taken over R.
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.
Build a bilinear interaction from elementary tensors
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Expand (e₁+2e₂)⊗(3f₁−f₂) in the basis eᵢ⊗fⱼ.
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
(e₁+2e₂)⊗w=e₁⊗w+2e₂⊗w
Bilinearity permits expansion in the first factor.
Work through the mathematics
=3e₁⊗f₁−e₁⊗f₂+6e₂⊗f₁−2e₂⊗f₂
Expand the second factor and collect the four coefficients.
Check the conclusion
The coefficient matrix is [[3,−1],[6,−2]], of rank one
A single elementary tensor corresponds to an outer product; sums of them need not have rank one.
The result
The coefficient matrix is [[3,−1],[6,−2]], of rank one
A single elementary tensor corresponds to an outer product; sums of them need not have rank one.
Common mistakes to catch
- A sum of elementary tensors need not remain elementary.
- Do not confuse tensor-product dimension with the sum of factor dimensions.
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
Is e₁⊗f₁+e₂⊗f₂ elementary in these two-dimensional spaces?
Show a hint
Look at the coefficient matrix rank.
Reveal answer and explanation
No
Its matrix is the identity, whose rank is two.
Practice 2
What is (2v)⊗w compared with v⊗(2w)?
Show a hint
Use bilinearity.
Reveal answer and explanation
They are equal
Both equal 2(v⊗w).
Take the idea with you
Describe a separable two-coordinate response as an outer-product matrix.
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: Check whether cosets support a group operation
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