Linear Algebra
Dynamical Geometric Problem Sheet 4: Linear Transforms in the Plane (Puzzles)
Instructions.
The following JSP-applet constructions contain several transforms in the Euclidean plane.
Here the plane is equipped with the standard vector space operations, the coordinate-wise
vector addition and multiplication by scalars.
You may find it convenient to look at the addition and multiplication by scalars separately -
use the appropriate Hide buttons.
Using the mouse you can
drag the points u and v, and
control the value of the real scalar c on the real line.
EXAMPLE: Translation by a constant vector in the plane
Problems (common to most puzzles below)
1. Is the function L linear?
2. What geometrical construction creates this function L ?
3. What symbolical formula stands behind L ?
4. In the case L is linear, find its matrix representation (rough approximation).
Hints
1. The function is not linear.
One can see it in many ways, the easiest is perhaps to note that
the zero vector is not mapped to itself.
One can, of course, use the definition of linearity, too.
2. You can easily see the translation vector a, but how?
3. L(u) := u + a.
1. First Puzzle
Problems for the First Puzzle
1. Is the function L linear?
2. What geometrical construction creates this function L ?
3. What symbolical formula stands behind L ?
4. In the case L is linear, find its matrix representation (rough approximation).
2. Second Puzzle
Problems for the Second Puzzle
1. Is the function L linear?
2. What geometrical construction creates this function L ?
3. What symbolical formula stands behind L ?
4. In the case L is linear, find its matrix representation (rough approximation).
3. Third Puzzle
Problems for the Third Puzzle
1. Is the function L linear?
2. What geometrical construction creates this function L ?
3. What symbolical formula stands behind L ?
4. In the case L is linear, find its matrix representation (rough approximation).
4. Fourth Puzzle
Problems for the Fourth Puzzle
1. Is the function L linear?
2. What geometrical construction creates this function L ?
3. Which hints did you use?
4. In the case L is linear, find its matrix representation (rough approximation).
5. Fifth Puzzle
Problems for the Fifth Puzzle
1. Is the function L linear?
2. What geometrical construction creates this function L ?
3. In the case L is linear, find its matrix representation (rough approximation).
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