nullspace of a matrix Isabel K Darcy Mathematics Department Applied Math and Computational Sciences University of Iowa Fig from knotplotcom Determine the column space of A Column space of A ID: 532096
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Slide1
A quick example calculating the column space and the nullspace of a matrix.
Isabel K. DarcyMathematics DepartmentApplied Math and Computational SciencesUniversity of Iowa
Fig from
knotplot.comSlide2
Determine the column space of A
=Slide3
Column space of A = span of the columns of A = set of all linear combinations of the columns of A Slide4
Column space of A = col A = col A = span , , ,
{
}
Determine the column space of A
=Slide5
Column space of A = col A = col A = span , , ,= c1
+ c2 +
c3 + c4
c
i
in R
{
}
{
}
Determine the column space of A
=Slide6
Column space of A = col A = col A = span , , ,= c1
+ c2 +
c3 + c4
c
i
in R
{
}
{
}
Determine the column space of A
=
Want simpler answerSlide7
Determine the column space of A
=
Put A into echelon form:
R
2
– R
1
R
2
R
3
+ 2R
1
R
3Slide8
Determine the column space of A
=
Put A into echelon form:
And determine the pivot columns
R
2
– R
1
R
2
R
3
+ 2R
1
R
3Slide9
Determine the column space of A
=
Put A into echelon form:
And determine the pivot columns
R
2
– R
1
R
2
R
3
+ 2R
1
R
3Slide10
Determine the column space of A
=
Put A into echelon form:
And determine the pivot columns
R
2
– R
1
R
2
R
3
+ 2R
1
R
3Slide11
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 =
R
2
– R
1
R
2
R
3
+ 2R
1
R
3
{
}Slide12
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.
{
}Slide13
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.
{
}Slide14
Determine the column space of A
=
{
}
col A contains all linear combinations of the 3 basis vectors:
col A = c
1
+
c
2
+
c
3
c
i
in R Slide15
Determine the column space of A
=
{
}
col A contains all linear combinations of the 3 basis vectors:
col A = c
1
+
c
2
+
c
3
c
i
in R
= span , ,
{
}Slide16
Determine the column space of A
=
{
}
col A contains all linear combinations of the 3 basis vectors:
col A = c
1
+
c
2
+
c
3
c
i
in R
= span , ,
{
}
Can you identify
col A?Slide17
Determine the nullspace of A
=
Put A into echelon form and then into reduced echelon
form:
R
2
– R
1
R
2
R
3
+ 2R
1
R
3
R
1
+
5R
2
R
1
R
2
/2 R
2
R
1
+ 8R
3
R
1
R
1
- 2R
3
R1
R3
/3 R3Slide18
Nullspace of A = solution set of Ax = 0Slide19
Solve: A x = 0
where A=
Put A
into echelon form and then into reduced echelon
form:
R
2
– R
1
R
2
R
3
+ 2R
1
R
3
R
1
+
5R
2
R
1
R
2
/2 R
2
R
1
+ 8R
3
R
1
R
1
- 2R3
R
1R
3/3 R3Slide20
Solve: A
x = 0 where A
~
x
1
x
2
x
3
x
4
0
0
0
x
1
-2
x
4
-
2
x
2
-
2
x
4
-
2
x
3
-
x
4
-1
x
4
x
4
1
=
=
x
4Slide21
Solve: A
x = 0 where A
~
x
1
x
2
x
3
x
4
0
0
0
x
1
-2
x
4
-
2
x
2
-
2
x
4
-
2
x
3
-
x
4
-1
x
4
x
4
1
=
=
x
4
Thus
Nullspace
of A =
Nul
A = x
4
x
4
in
R
{
}Slide22
Solve: A
x = 0 where A
~
x
1
x
2
x
3
x
4
0
0
0
Thus
Nullspace
of A =
Nul
A = x
4
x
4
in
R
= span
{
}
{
}