Practice question · Multiple choice
Rotation and scaling are linear transformations of the plane; translation is not. Why does moving every point by a fixed offset fail to qualify?
Hints
- Apply the scaling rule with c = 0. What does it force T(0) to be?
- Where does a translation send the origin?
Show the answer
A. Because a linear transformation must fix the origin, and translation moves it.
Why
Linearity requires T(cv) = cT(v), and c = 0 forces T(0) = 0, rotation and scaling fix the origin, translation by definition does not. Note translation preserves all distances exactly; its failure is about the origin, not distortion. And the homogeneous-coordinate trick embeds the plane in three dimensions where translation becomes a shear, which is linear.
Practise Linear Transformations
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More questions on Linear Transformations
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