let S={v1,v2,v3} subset R3. check for the following choice of v1,v2,v3 and w whether w exist span(S). v1=(3,4,7), v2=(1,0,3), v3=(2,6,3), w=(3,1,1) v1,v2,v3,w are vectors

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W is in Span(S) iff w is a linear combination of `v_1 v_2 v_3`

iff there exists a,b,c such that `w=av_1+bv_2+cv_3`

iff considering the coordinates,

 

`3=3a+b+2c `

`4a+0b+6c=1 `

`7a+3b+3c=1`

 

 

iff

`b=3-3a-2c `

`4a+6c=1 `

`7a+3(3-3a-2c)+3c=1`

 


iff

`b=3-3a-2c `

`4a+6c=1 `

`7a+9-9a-6c+3c=1`

 

iff

`b=3-3a-2c `

`4a+6c=1 `

`2a+3c=8 lt=gt 4a+6c=16` (multiply the inequality by 2)

There are not such solution since 4a+6c can't be at the same time equal to 1 and 16

Therefore w doesn't belong to `span(v_1,v_2,v_3)`

 

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