how to determine whether a given set of vectors spans r 3
DESCRIPTION
How to determine whether a given set of vectors SPANS R 3. example 1: This set does SPAN R 3. example 2: This set does NOT SPAN R 3. Do solutions exist for all x, y, and z ?. Do solutions exist for all x, y, and z ?. Do solutions exist for all x, y, and z ?. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/1.jpg)
How to determine whether a given set of vectorsSPANS R3
example 1:This set does SPAN R 3
example 2:This set does NOT SPAN R 3
![Page 2: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/2.jpg)
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes 3
![Page 3: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/3.jpg)
:such that ,, scalars thereare any For
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes
321
3
ccc
z
y
x
![Page 4: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/4.jpg)
321
321
321
321
321
3
211
321
432
2
3
4
1
2
3
1
1
2
:such that ,, scalars thereare any For
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes
cccz
cccy
cccx
ccc
z
y
x
ccc
z
y
x
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
![Page 5: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/5.jpg)
321
321
321
321
321
3
211
321
432
2
3
4
1
2
3
1
1
2
:such that ,, scalars thereare any For
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes
cccz
cccy
cccx
ccc
z
y
x
ccc
z
y
x
![Page 6: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/6.jpg)
321
321
321
321
321
3
211
321
432
2
3
4
1
2
3
1
1
2
:such that ,, scalars thereare any For
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes
cccz
cccy
cccx
ccc
z
y
x
ccc
z
y
x
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
![Page 7: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/7.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
![Page 8: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/8.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
![Page 9: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/9.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
![Page 10: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/10.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
![Page 11: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/11.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
-1
0 1 2 y-(x-y)
)(100
321
111
yxz
y
yx
![Page 12: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/12.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
-1
0 1 2 y-(x-y)
)(100
321
111
yxz
y
yx
-1
0 -1 (x-y)-(2y-x)
)(
2
100
210
111
yxz
xy
yx
![Page 13: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/13.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
-1
0 1 2 y-(x-y)
)(100
321
111
yxz
y
yx
-1
0 -1 (x-y)-(2y-x)
)(
2
100
210
111
yxz
xy
yx
)(
2
32
100
210
101
yxz
xy
yx
+1
0 (2x-3y)+(z-(x-y))
![Page 14: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/14.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
-1
0 1 2 y-(x-y)
)(100
321
111
yxz
y
yx
-1
0 -1 (x-y)-(2y-x)
)(
2
100
210
111
yxz
xy
yx
)(
2
2
100
210
001
yxz
xy
zyx
-2
0 (2y-x)-2(z-(x-y))
![Page 15: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/15.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
-1
0 1 2 y-(x-y)
)(100
321
111
yxz
y
yx
-1
0 -1 (x-y)-(2y-x)
)(
2
100
210
111
yxz
xy
yx
)(
2
2
100
010
001
yxz
zx
zyx
![Page 16: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/16.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
211
321
432
:reduce
z
y
x
211
321
432
-1
1 1 1 x-y
z
y
yx
211
321
111-1
0 0 1 z-(x-y)
-1
0 1 2 y-(x-y)
)(100
321
111
yxz
y
yx
-1
0 -1 (x-y)-(2y-x)
)(
2
100
210
111
yxz
xy
yx
zyx
zx
zyx
2
2
100
010
001
YES
c1
c2
c3
=
=
=
![Page 17: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/17.jpg)
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
YES
c1
c2
c3
=
=
=
zyx
zx
zyx
2
2
100
010
001
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes 3
YES
return to outline
![Page 18: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/18.jpg)
?R SPAN
1
3
4
,
1
2
3
,
1
1
2
set theDoes 3
![Page 19: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/19.jpg)
:such that ,, scalars thereare any For
?R SPAN
1
3
4
,
1
2
3
,
1
1
2
set theDoes
321
3
ccc
z
y
x
![Page 20: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/20.jpg)
321
321
321
321
321
3
111
321
432
1
3
4
1
2
3
1
1
2
:such that ,, scalars thereare any For
?R SPAN
1
3
4
,
1
2
3
,
1
1
2
set theDoes
cccz
cccy
cccx
ccc
z
y
x
ccc
z
y
x
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
![Page 21: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/21.jpg)
321
321
321
321
321
3
111
321
432
1
3
4
1
2
3
1
1
2
:such that ,, scalars thereare any For
?R SPAN
1
3
4
,
1
2
3
,
1
1
2
set theDoes
cccz
cccy
cccx
ccc
z
y
x
ccc
z
y
x
![Page 22: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/22.jpg)
321
321
321
321
321
3
111
321
432
1
3
4
1
2
3
1
1
2
:such that ,, scalars thereare any For
?R SPAN
1
3
4
,
1
2
3
,
1
1
2
set theDoes
cccz
cccy
cccx
ccc
z
y
x
ccc
z
y
x
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
![Page 23: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/23.jpg)
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
![Page 24: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/24.jpg)
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
111
321
432
:reduce
![Page 25: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/25.jpg)
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
111
321
432
:reduce
z
y
x
111
321
432
-1
1 1 1 x-y
![Page 26: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/26.jpg)
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
111
321
432
:reduce
z
y
x
111
321
432
-1
1 1 1 x-y
z
y
yx
111
321
111-1
0 0 0 z-(x-y)
![Page 27: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/27.jpg)
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
111
321
432
:reduce
z
y
x
111
321
432
-1
1 1 1 x-y
z
y
yx
111
321
111-1
0 0 0 z-(x-y)
)(000
321
111
yxz
y
yx
There are no solutions if z – x + y does not equal zero
![Page 28: How to determine whether a given set of vectors SPANS R 3](https://reader036.vdocuments.us/reader036/viewer/2022062323/56815a9b550346895dc81970/html5/thumbnails/28.jpg)
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
z
y
x
111
321
432
:reduce
z
y
x
111
321
432
-1
1 1 1 x-y
z
y
yx
111
321
111-1
0 0 0 z-(x-y)
)(000
321
111
yxz
y
yx
There are no solutions if z – x + y does not equal zero
1
3
4
1
2
3
,
1
1
2
ofn combinatiolinear anot is
3
1
1
The set doesNot span R3