Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compose an affine map using a matrix and a displacement.
- Justify the conclusion "T(0)=b≠0, so T is affine but not linear" using the stated assumptions.
Before you start
Matrix multiplication and vectors.
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 T(x)=Ax+b with A=[[2,0],[0,1]], b=(1,−2), find T(3,4) and test linearity.
Why this math matters
Compose an affine map using a matrix and a displacement. 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:
- A and b are fixed.
- The stretching matrix is invertible.
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 translation from a linear transformation
PausedQuestion: Start with the question. Paused.
Question
Start with the question
For T(x)=Ax+b with A=[[2,0],[0,1]], b=(1,−2), find T(3,4) and test linearity.
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(3,4)=(6,4)
The linear part stretches only the first coordinate.
Work through the mathematics
T(3,4)=(6,4)+(1,−2)=(7,2)
Translation is applied after the matrix action.
Check the conclusion
T(0)=b≠0, so T is affine but not linear
Linear maps fix zero; this transformation preserves affine combinations instead.
The result
T(0)=b≠0, so T is affine but not linear
Linear maps fix zero; this transformation preserves affine combinations instead.
Common mistakes to catch
- Translation cannot be represented by a 2×2 linear map on R² alone.
- Changing operation order changes the displacement.
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
Compute T((p+q)/2).
Show a hint
Distribute A and account for b once.
Reveal answer and explanation
(T(p)+T(q))/2
Midpoints are preserved by affine maps.
Practice 2
What is the inverse formula?
Show a hint
Undo translation before stretching.
Reveal answer and explanation
T⁻¹(y)=A⁻¹(y−b)
Reversing the operations in reverse order recovers the input.
Take the idea with you
Describe an image resize followed by a screen-position shift.
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: Locate a point through triangle weights
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