MathPlotAI Logo MathPlotAI

The Four Subspaces of a Matrix

The Four Subspaces of a Matrix
Loading the math board and drawing, please wait…
2014-09-02 08:59
VectorMatrixGeometric Construction
In collections Complex Math
This example demonstrates the four subspaces of a matrix (column space, row space, null space, left null space) through visualizations. Key math concepts include subspaces, basis, and dimension in linear algebra. Constructed using GeoGebra commands like Segment, Vector, Circle, and involves numerous objects such as text and vectors. Suitable for high school students and beginners in linear algebra.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(A_1, B_1) → a_1
  3. Segment(C, D) → b
  4. Segment(C_1, D_1) → b_1
  5. Segment(A_2, B_2) → a_2
  6. Segment(A_3, B_3) → a_3
  7. Segment(C_2, D_2) → b_2
  8. Segment(C_3, D_3) → b_3
  9. Segment(C_4, D_4) → b_4
  10. Segment(C_5, D_5) → b_5
  11. Segment(A_4, B_4) → a_4
  12. Segment(A_5, B_5) → a_5
  13. Segment(A_6, B_6) → a_6
  14. Segment(A_7, B_7) → a_7
  15. Segment(A_8, B_8) → a_8
  16. Segment(C_6, D_6) → b_6
  17. Segment(C_7, D_7) → b_7
  18. Segment(E, F) → c
  19. Segment(E_1, F_1) → c_1
  20. Segment(E_2, F_2) → c_2
  21. Segment(E_3, F_3) → c_3
  22. Segment(E_4, F_4) → c_4
  23. Segment(E_5, F_5) → c_5
  24. Segment(G, H) → d
  25. Segment(G_1, H_1) → d_1
  26. Vector(I, J) → u
  27. Segment(E_6, F_6) → c_6
  28. Segment(E_7, F_7) → c_7
  29. Segment(K, L) → e
  30. Segment(K_1, L_1) → e_1
  31. Segment(M, N) → f
  32. Segment(K_2, L_2) → e_2
  33. Segment(K_3, L_3) → e_3
  34. Segment(K_4, L_4) → e_4
  35. Segment(E_8, F_8) → c_8
  36. Circle(O, P) → g
  37. Circle(O_1, P_1) → g_1
  38. Segment(C_8, D_8) → b_8
  39. Segment(C_9, D_9) → b_9
  40. Segment(C_{10}, D_{10}) → b_{10}
  41. Segment(C_{11}, D_{11}) → b_{11}
  42. Segment(C_{12}, D_{12}) → b_{12}
  43. Segment(C_{13}, D_{13}) → b_{13}
  44. Segment(C_{14}, D_{14}) → b_{14}
  45. Segment(C_{15}, D_{15}) → b_{15}
  46. Segment(Q, R) → h
  47. Segment(Q_1, R_1) → h_1
  48. Segment(Q_2, R_2) → h_2
  49. Segment(Q_3, R_3) → h_3
  50. Segment(E_9, F_9) → c_9
  51. Segment(E_{10}, F_{10}) → c_{10}
  52. Segment(E_{11}, F_{11}) → c_{11}
  53. Segment(E_{12}, F_{12}) → c_{12}
  54. Segment(E_{13}, F_{13}) → c_{13}
  55. Segment(E_{14}, F_{14}) → c_{14}
  56. Segment(E_{15}, F_{15}) → c_{15}
  57. Segment(E_{16}, F_{16}) → c_{16}

AI Drawing Prompt Reference

Visualize the four subspaces of a matrix: draw line segments for the column and row spaces, vectors for basis vectors, and circles to represent geometric relationships in higher dimensions, labeling each subspace. Start by sketching parallel line segments to depict different subspaces, then connect key points with vectors, and finally use circles to highlight higher-dimensional properties, creating a clear illustration of linear algebra concepts like subspaces and basis.
The prompt above was compiled from the original drawing commands and is for reference only.