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.

Expert Answers
rakesh05 eNotes educator| Certified Educator

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]}.

pramodpandey | Student

 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  



Write row echolon form of coefficient matrix ,we have



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.