Let W be a subspace of a finite dimensional vector space a S

Let W be a subspace of a finite dimensional vector space a. Say that dim(V)=n, dim(W)=k and choose a basis T=[v1,v2...vk] for W. Show that each integer, m, in the following set satisfied k<=m<=n The set: P=[m(integer)|m=|Beta| for some linearly independent set Beta statisying T is a subset of Beta is a subset of V. b. Prove that any basis T of W can be extended to a basis Beta of V.

Solution

T = [v1,v2,v3, ..............,vk] is a basis for W so T spans W vi\'s are linearly independant for 1<= i <= k .

now take a out side element of W in V say a1 so v1,v2,v3, ....................,vk,a1 are linearly independent as a1 does not belongs to span of v1,v2,v3,.........,vk ie T. let\'s call the set spanned by v1,v2,v3.......,vk,a1 be T1.

now T1 is a subspace of V as v1,v2,............,vk,a1 belongs to V.

T1 has k+1 elemants and formed from a arbitrary basis of T. now by carring the same process we can get k+1 to k+2 elements and so on till n. because the process cannot end before n as every basis of a vector space contains equal elements. so if it end before n say in Tx then we Tx has to span V otherwise we can find a elemant in V which is not in span of Tx. so contradiction.

CONCLUTION : any basis T of W can be extend any dimentional(>=k &<= n) subspace of V . and any basis of T can be extended to a basis of a subspace of V which contains W.