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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Solve for coordinates relative to a nonstandard basis.
- Justify the conclusion "[v]B=(3,1), since 3(1,1)+(1,−1)=(4,2)" using the stated assumptions.
Before you start
Vector addition and invertible matrices.
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
Write v=(4,2) in the basis b₁=(1,1), b₂=(1,−1).
Why this math matters
Solve for coordinates relative to a nonstandard 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 basis vectors are ordered as stated.
- Coordinates are relative to that order.
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.
Describe the same vector in another coordinate system
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Write v=(4,2) in the basis b₁=(1,1), b₂=(1,−1).
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
v=c₁b₁+c₂b₂ gives c₁+c₂=4, c₁−c₂=2
Coordinates are coefficients of basis vectors, not necessarily the usual components.
Work through the mathematics
2c₁=6 ⇒ c₁=3; c₂=1
Adding the equations isolates the first coefficient.
Check the conclusion
[v]B=(3,1), since 3(1,1)+(1,−1)=(4,2)
The vector is unchanged; only its coordinate description has changed.
The result
[v]B=(3,1), since 3(1,1)+(1,−1)=(4,2)
The vector is unchanged; only its coordinate description has changed.
Common mistakes to catch
- Do not identify a vector's coordinates with the vector itself.
- Swapping basis order swaps coordinate positions.
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
Find the B-coordinates of (2,4).
Show a hint
Add and subtract the component equations.
Reveal answer and explanation
(3,−1)
3(1,1)−(1,−1)=(2,4).
Practice 2
Can dependent vectors give unique coordinates for every plane vector?
Show a hint
Consider an invertible basis matrix.
Reveal answer and explanation
No
Dependence makes the matrix singular and prevents a unique representation of every vector.
Take the idea with you
Convert a motion vector into forward and sideways components along rotated axes.
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: Turn differentiation into a matrix
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