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Column Space A: Visualizing Vector Spaces

Column Space A: Visualizing Vector Spaces
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2022-11-17 14:55
VectorCoordinate GeometryMatrix
This example illustrates the concept of column space A, which is the space generated by the column vectors of matrix A. Using GeoGebra commands such as Vector, Intersect, and PointIn, the geometric meaning of vector spaces is visually demonstrated. Suitable for teaching vector spaces and linear combinations in linear algebra courses.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, zAxis) → A
  2. PointIn(xOyPlane) → B
  3. Vector(A, B) → u
  4. Vector(A, C) → v
  5. Vector(A, D) → w
  6. Vector(C, D) → a
  7. Angle(D, C, A) → α
  8. Vector(B, D) → b
  9. Segment(B, D) → f
  10. Segment(C, D) → g
  11. LaTeX(err1, true, true) → text1
  12. Segment((0, 0, 0), C) → h
  13. Surface(u_{1}, v_{1}, 0, u_{1}, -5, 5, v_{1}, -5, 5) → c
  14. Text("$Col(\textit{A})$", E, false, true) → text3
  15. Point(h) → G
  16. Text("$\textit{A}\hat{\mathbf x}=\text{Proj}_{Col(\textit{A})}(\mathbf b)$", G + (-0, -0, 0.35), false, true) → text4

AI Drawing Prompt Reference

Visualize the column space in 3D by plotting the intersection of the x-axis and z-axis, then draw three vectors originating from this point to two points on the xy-plane and one point in space. Connect the two points on the plane with a line segment. Draw a vector from the origin to the space point and generate a rectangular surface using two of the vectors. Label the column space on the surface and draw a vector from the origin to the space point, indicating the projection relation.
The prompt above was compiled from the original drawing commands and is for reference only.