Let B = {v 1 ,… v k }an independent set of vectors in a vector
space V , and u V . Prove that if u span(B) then the set {v 1 ,… v k ,u}is
independent.
Solution
Let a 1 v 1 + a 2 v 2 + .... + a k v k + bu = 0 ---------------(1)
Now if b not equal to 0
then u = (-1/b)(a 1 v 1 + a 2 v 2 + .... + a k v k )
=> u span(B)
But given u span(B)
=> b=0
=> a 1 v 1 + a 2 v 2 + .... + a k v k = 0
But given B = {v 1 ,… v k } is independent set
=> a 1 = a 2 = .... = a k = 0
Hence  a 1 = a 2 = .... = a k = b = 0
=> the set {v 1 ,… v k ,u} is independent.
.
Let B - {v1- vk }an independent set of vectors in a vector space V - a.docx
1. Let B = {v 1 ,… v k }an independent set of vectors in a vector
space V , and u V . Prove that if u span(B) then the set {v 1 ,… v k ,u}is
independent.
Solution
Let a 1 v 1 + a 2 v 2 + .... + a k v k + bu = 0 ---------------(1)
Now if b not equal to 0
then u = (-1/b)(a 1 v 1 + a 2 v 2 + .... + a k v k )
=> u span(B)
But given u span(B)
=> b=0
=> a 1 v 1 + a 2 v 2 + .... + a k v k = 0
But given B = {v 1 ,… v k } is independent set
=> a 1 = a 2 = .... = a k = 0
Hence  a 1 = a 2 = .... = a k = b = 0
=> the set {v 1 ,… v k ,u} is independent.
