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A quick example calculating the column space and the nullspace of a matrix.

This guide illustrates how to calculate the column space and null space of a matrix using echelon forms. It defines the column space as the span of its columns, derived from all linear combinations of the columns of the matrix A. The example provides steps to determine the pivot columns and form a basis for the column space. Furthermore, it discusses finding the null space by converting the matrix into echelon form and then to reduced echelon form, ultimately solving the equation Ax = 0.

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A quick example calculating the column space and the nullspace of a matrix.

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  1. A quick example calculating the column space and the nullspace of a matrix. Isabel K. Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplot.com

  2. = Determine the column space of A

  3. Column space of A = span of the columns of A = set of all linear combinations of the columns of A

  4. = Determine the column space of A Column space of A = col A = col A = span , , , { }

  5. = Determine the column space of A Column space of A = col A = col A = span , , , = c1+ c2+c3 + c4 ciin R { } { }

  6. = Determine the column space of A Column space of A = col A = col A = span , , , = c1+ c2+c3 + c4 ciin R { } Want simpler answer { }

  7. = Determine the column space of A Put A into echelon form: R2 – R1 R2 R3 + 2R1  R3

  8. = Determine the column space of A Put A into echelon form: And determine the pivot columns R2 – R1 R2 R3 + 2R1  R3

  9. = Determine the column space of A Put A into echelon form: And determine the pivot columns R2 – R1 R2 R3 + 2R1  R3

  10. = Determine the column space of A Put A into echelon form: And determine the pivot columns R2 – R1 R2 R3 + 2R1  R3

  11. = Determine the column space of A Put A into echelon form: A basis for col A consists of the 3 pivot columns from the original matrix A. Thus basis for col A = R2 – R1 R2 R3 + 2R1  R3 { }

  12. = Determine the column space of A A basis for col A consists of the 3 pivot columns from the original matrix A. Thus basis for col A = Note the basis for col A consists of exactly 3 vectors. { }

  13. = Determine the column space of A A basis for col A consists of the 3 pivot columns from the original matrix A. Thus basis for col A = Note the basis for col A consists of exactly 3 vectors. Thus col A is 3-dimensional. { }

  14. = Determine the column space of A col A contains all linear combinations of the 3 basis vectors: col A = c1+ c2+ c3 ci in R { }

  15. = Determine the column space of A col A contains all linear combinations of the 3 basis vectors: col A = c1+ c2+ c3 ci in R = span , , { } { }

  16. = Determine the column space of A col A contains all linear combinations of the 3 basis vectors: col A = c1+ c2+ c3 ci in R = span , , { } { } Can you identify col A?

  17. = Determine the nullspace of A Put A into echelon form and then into reduced echelon form: R2 – R1 R2 R3 + 2R1  R3 R1+ 5R2 R1 R2/2  R2 R1 + 8R3 R1 R1- 2R3 R1 R3/3  R3

  18. Nullspace of A = solution set of Ax = 0

  19. = Solve: A x = 0 where A Put A into echelon form and then into reduced echelon form: R2 – R1 R2 R3 + 2R1  R3 R1+ 5R2 R1 R2/2  R2 R1 + 8R3 R1 R1- 2R3 R1 R3/3  R3

  20. 0 0 0 ~ Solve: A x = 0 where A x1 x2x3x4 x1 -2x4-2 x2 -2x4-2 x3-x4-1 x4x41 = = x4

  21. 0 0 0 ~ Solve: A x = 0 where A x1 x2x3x4 x1 -2x4-2 x2 -2x4-2 x3-x4-1 x4x41 = = { Thus Nullspace of A = Nul A = x4 x4 in R } x4

  22. 0 0 0 ~ Solve: A x = 0 where A x1 x2x3x4 Thus Nullspace of A = Nul A = x4 x4 in R = span { } { }

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