a quick example calculating the column space and the nullspace of a matrix
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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. =. Determine the column space of A . Column space of A - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/1.jpg)
A quick example calculating the column space and the
nullspace of a matrix.
Isabel K. DarcyMathematics DepartmentApplied Math and Computational SciencesUniversity of IowaFig from
knotplot.com
![Page 2: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/2.jpg)
Determine the column space of A =
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Column space of A
= span of the columns of A
= set of all linear combinations
of the columns of A
![Page 4: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/4.jpg)
Column space of A = col A = col A = span , , ,
{ }Determine the column space of A =
![Page 5: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/5.jpg)
Column space of A = col A = col A = span , , ,
= c1 + c2 + c3 + c4 ci in R { }
{ }Determine the column space of A =
![Page 6: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/6.jpg)
Column space of A = col A = col A = span , , ,
= c1 + c2 + c3 + c4 ci in R { }
{ }Determine the column space of A =
Want simpler answer
![Page 7: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/7.jpg)
Determine the column space of A =
Put A into echelon form:
R2 – R1 R2
R3 + 2R1 R3
![Page 8: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/8.jpg)
Determine the column space of A =
Put A into echelon form:
And determine the pivot columns
R2 – R1 R2
R3 + 2R1 R3
![Page 9: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/9.jpg)
Determine the column space of A =
Put A into echelon form:
And determine the pivot columns
R2 – R1 R2
R3 + 2R1 R3
![Page 10: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/10.jpg)
Determine the column space of A =
Put A into echelon form:
And determine the pivot columns
R2 – R1 R2
R3 + 2R1 R3
![Page 11: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/11.jpg)
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
{ }
![Page 12: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/12.jpg)
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.
{ }
![Page 13: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/13.jpg)
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.
{ }
![Page 14: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/14.jpg)
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
![Page 15: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/15.jpg)
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 , , { }
![Page 16: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/16.jpg)
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?
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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
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Nullspace of A = solution set of Ax = 0
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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
![Page 20: A quick example calculating the column space and the nullspace of a matrix](https://reader030.vdocument.in/reader030/viewer/2022020308/56815f61550346895dce48e2/html5/thumbnails/20.jpg)
Solve: A x = 0 where A ~x1 x2 x3 x4
000
x1 -2 x4 -2
x2 -2 x4 -2
x3 - x4 -1x4 x4 1
= = x4
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Solve: A x = 0 where A ~x1 x2 x3 x4
000
x1 -2 x4 -2
x2 -2 x4 -2
x3 - x4 -1x4 x4 1
= = x4Thus Nullspace of A =
Nul A = x4 x4 in R { }
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Solve: A x = 0 where A ~x1 x2 x3 x4
000
Thus Nullspace of A =
Nul A = x4 x4 in R = span{ } {}