# Consider the set S = { <1,-2,2,5> , <4,-5,6,-2> , <-7,5,-8,31> , <-1,-1,2,3> Find conditions on <a, b, c, d>to be in SpanS and find the basis for the span of S.

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If we prepare the matrix corresponding to these vectors we get

`[[1,-2,2,5],[4,-5,6,-2],[-7,5,-8,31],[-1,-1,2,3]]` `~~` `[[1,-2,2,5],[0,-3,2,22],[0,-9,6,66],[0,-3,4,8]]` `~~` `[[1,-2,2,5],[0,-3,2,22],[0,0,0,0],[0,0,-2,14]]`

`~~` `[[1,-2,2,5],[0,-3,2,22],[0,0,-2,14],[0,0,0,0]]` `~~` `[[1,-2,2,5],[0,1,-2/3,-22/3],[0,0,1,-7],[0,0,0,0]]`

Because S is a subset of R^4. So rank of the matrix should be 4. But here we see that rank of the matrix is 3. So The condition for [a,b,c,d] to be in span S is that a=0, b=0,c=0 and d is any non-zero real number.

Basis for span S is {[1,-2,2,5],[0,1,-2/3,-22/3],[0,0,1,-7],[0,0,0,1]}.

Conditions on <a, b, c, d>to be Span S is that vectors a ,b c,and d to be linearly independent.

let w,x,y z be real nos. sucht that

w<1,-2,2,5>+x<4,-5,6,-2>+y<-7,5,-8,-31>+z<-1,-1,2,3>=<0,0,0,0> then

w=x=y=z=0

`[[1,4,-7,-1],[-2,-5,5,-1],[2,6,8,2],[5,-2,31,3]][[w],[x],[y],[z]]=[[0],[0],[0],[0]]`

Write row echolon form of coefficient matrix ,we have

`[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]][[w],[x],[y],[z]]=[[0],[0],[0],[0]]`

x=y=z=w=0

so vectors <1,-2,2,5>,<4,-5,6,-2>,<-7,5,-8,-31>,<-1,-1,2,3> are linearly independen.hence basis of S.