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Change of Basis by Pablo Rodríguez-Sánchez

Change of Basis by Pablo Rodríguez-Sánchez
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2015-03-15 19:43
VectorMatrixTransformation
This example illustrates the concept of change of basis in linear algebra. By inputting vectors and bases, it visually demonstrates how vectors are represented differently under various bases. Commands like Vector, Segment, Line, and Intersect are used, suitable for linear algebra courses.

GeoGebra Commands & Parameters

  1. Textfield(a_x) → inputBox1
  2. Textfield(a_y) → inputBox2
  3. Textfield(b_x) → inputBox3
  4. Textfield(b_y) → inputBox4
  5. Vector((a_x, a_y)) → a
  6. Vector((b_x, b_y)) → b
  7. Vector(R) → r
  8. Segment(R, (x(R), 0)) → SegmentXBc
  9. Segment(R, (0, y(R))) → SegmentYBc
  10. Line((0, 0), a) → AxisXB
  11. Line((0, 0), b) → AxisYB
  12. Line(R, a) → LineXB
  13. Line(R, b) → LineYB
  14. Intersect(AxisXB, LineYB) → AuxXB
  15. Intersect(AxisYB, LineXB) → AuxYB
  16. Segment(AuxXB, R) → SegmentXB
  17. Segment(AuxYB, R) → SegmentYB
  18. Determinant(MatrixBtoBc) → det
  19. Invert(MatrixBtoBc) → MatrixBctoB
  20. Sequence(Line[(t, 0), (t, 1)], t, (-TicksMax), TicksMax, 1) → GridXBc
  21. Sequence(Line[(0, t), (1, t)], t, (-TicksMax), TicksMax, 1) → GridYBc
  22. Sequence(Point[(0, 0), Vector[(a * t)]], t, (-TicksMax), TicksMax, 1) → TickMarksXB
  23. Sequence(Point[(0, 0), Vector[(b * t)]], t, (-TicksMax), TicksMax, 1) → TickMarksYB
  24. Sequence(Line[Point[(0, 0), Vector[(a * t)]], b], t, (-TicksMax), TicksMax, 1) → GridXB
  25. Sequence(Line[Point[(0, 0), Vector[(b * t)]], a], t, (-TicksMax), TicksMax, 1) → GridYB

AI Drawing Prompt Reference

Draw two basis vectors in a coordinate system and visualize how the same vector is represented differently under various bases. Start by plotting the axes defined by the basis vectors, then show the transformation of a vector's coordinates under different bases. Include grid lines and tick marks to clearly illustrate the differences in representation, suitable for linear algebra education.
The prompt above was compiled from the original drawing commands and is for reference only.