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2D and 1D Eigenspaces

2D and 1D Eigenspaces
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2020-04-30 02:40
VectorMatrixTransformationGeometric Construction
This example demonstrates the geometric behavior of 2D and 1D eigenspaces under matrix transformations, involving core concepts like eigenvalues, eigenvectors, matrix operations, and linear algebra. It constructs transformation matrices T, identity matrix I, matrix A, and eigenvectors e_{1}, e_{2}, e_{3} using GeoGebra commands ApplyMatrix and Sequence, with sliders t_{1}, t_{2}, t_{3} for interactive exploration. Ideal for university-level linear algebra or advanced mathematics instruction.

GeoGebra Commands & Parameters

  1. ApplyMatrix(M, e_{1}) → t_{1}
  2. ApplyMatrix(M, e_{2}) → t_{2}
  3. ApplyMatrix(M, e_{3}) → t_{3}
  4. Sequence(Sequence[Sequence[(x_{1}, x_{2}, x_{3}), x_{1}, -4, 4, 1], x_{2}, -4, 4, 1], x_{3}, -4, 4, 1) → l1
  5. ApplyMatrix(M, l1) → l1'
  6. ApplyMatrix(M, x_{1}) → u
  7. ApplyMatrix(M, x_{2}) → v
  8. ApplyMatrix(M, x_{3}) → w
  9. Sequence(ApplyMatrix[M, (t, t, t)], t, -5, 5, 0.2) → l2
  10. Sequence(Sequence[Point[(0, 0, 0), Vector[(m * v) + (n * w)]], m, -5, 5], n, -5, 5) → l5
  11. ApplyMatrix(M, l5) → l5'

AI Drawing Prompt Reference

Draw the geometric behavior of 2D and 1D eigenspaces under matrix transformations. First, create a matrix T, then construct eigenvectors e1, e2, and e3. Next, apply matrix T to these vectors to observe their transformations. Finally, show the transformation effects of eigenspaces by applying matrix T, including the transformations of eigenvectors e1, e2, and e3 and the lines and points they generate.
The prompt above was compiled from the original drawing commands and is for reference only.