to determine whether a given set of vec SPANS R 3 example 1: This set does SPAN R 3 example 2: This set does NOT SPAN R
How to determine whether a given set of vectors SPANS R 3
Feb 06, 2016
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 Presentation
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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
?R SPAN
2
3
4
,
1
2
3
,
1
1
2
set theDoes 3
: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
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 ?
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
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 ?
3
2
1
211
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
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
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
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)
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
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
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))
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))
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
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
=
=
=
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
?R SPAN
1
3
4
,
1
2
3
,
1
1
2
set theDoes 3
: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
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 ?
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
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 ?
3
2
1
111
321
432
c
c
c
z
y
x
Do solutions exist for all x, y, and z ?
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
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
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)
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
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
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