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Spaces Associated with a Matrix

Spaces Associated with a Matrix
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2017-02-24 08:04
VectorCoordinate GeometryMatrix
In collections Linear Algebra
This example demonstrates the null space, column space, and related geometric representations of a matrix. Core mathematical concepts include rank, linear independence, and linear transformations. GeoGebra commands like Random, Sequence, Sort, ReducedRowEchelonForm, Identity, Remove, and Transpose are used in construction. Suitable for high school students and beginners in linear algebra learning matrix spaces.

GeoGebra Commands & Parameters

  1. Random(1, 3) → nullity
  2. Random(2, 4) → s
  3. Sequence(i, i, 1, n) → olist
  4. Sort(Take[Shuffle[olist], n - s + 1]) → olistcut
  5. Sequence(Sequence[If[i ≟ Element[olistcut, j], 1, i > Element[olistcut, j], Random[-5, 5], 0], i, 1, n], j, 1, n) → ns
  6. ReducedRowEchelonForm(ns) → rrofns
  7. Random(1, nullity) → npad
  8. Random(1, nrow) → s2
  9. Random(1, nrow) → s1
  10. Sequence(Sequence[If[(i ≟ j ∧ i ≟ s1 ∨ i ≟ j ∧ i ≟ s2) ∧ (s1 ≠ s2), -1, 0], i, 1, nrow], j, 1, nrow) → swaptozero
  11. Sequence(Sequence[If[i ≟ s1 ∧ j ≟ s2 ∨ i ≟ s2 ∧ j ≟ s1, 1, If[i ≟ j, 1, 0]], i, 1, nrow], j, 1, nrow) → startswap
  12. Sequence(Sequence[Element[rrofns, i, j], j, 1, n], i, 1, s + npad) → notsquare
  13. Sequence(Sequence[If[i > j, Random[-5, 5], 0], i, 1, s + npad], j, 1, s + npad) → t5
  14. Sequence(Sequence[If[i < j, Random[-5, 5], 0], i, 1, s + npad], j, 1, s + npad) → t6
  15. Identity(s + npad) → idenpad
  16. Remove(olist, olistcut) → par
  17. Sequence(Sequence[If[i ≟ Element[par, j], 1, IndexOf[i, par]^(2) > 0, 0, (-Element[rrofns, IndexOf[i, olistcut], Element[par, j]])], i, 1, n], j, 1, n - s) → basisnull
  18. Sequence(Sequence[Element[rrofns, i, j], j, 1, n], i, 1, s) → rightsize
  19. Identity(s) → iden
  20. Sequence(Sequence[If[i < j, Random[-5, 5], 0], i, 1, s], j, 1, s) → t1
  21. Sequence(Sequence[If[i > j, Random[-5, 5], 0], i, 1, s], j, 1, s) → t2
  22. ReducedRowEchelonForm(togive) → togivecheck
  23. Transpose(togivenotsquare) → transposespace
  24. ReducedRowEchelonForm(transposespace) → transposespacerref
  25. Sequence(Sequence[If[i < j, Random[-5, 5], 0], i, 1, n], j, 1, n) → t3
  26. Identity(n) → idenn
  27. Sequence(Sequence[If[i > j, Random[-5, 5], 0], i, 1, n], j, 1, n) → t4
  28. ReducedRowEchelonForm(togivesquare) → togivesquarecheck
  29. Transpose(togivesquare) → transposetogivesquare
  30. ReducedRowEchelonForm(transposetogivesquare) → transpsquaretogive
  31. ReducedRowEchelonForm(togivenotsquare) → togivenotsquarecheck