Let B be a basis of an mdimensional subspace W of Rn Prove t

Let B be a basis of an m-dimensional subspace W of R^n. Prove that a set {v_1, ..., v_k} of vectors in W is linearly independent in R^n if and only if {(v_1)B, ..., (v_k)B} is linearly independent in R^m.

Solution

Given that B is basis of W contained in Rⁿ

and dim(W)= m

let B ={v₁,v₂,v_3...v_m}

and {v₁,v₂,...v_k} is a set in W for m<k

we have B is basis of W

so, 1) B is L.I

   2)L(B)=W

B is L.I so

{v₁,v_2,v₃...v_m} is L.I

so

a₁v₁+a₂v₂+....,+a_mv_m=0

a₁=a₂=....=a_m=0

consider a₁(v₁)_B+a₂(v₂)_B+...a_k(v_k)_B=0

a₁(v₁)_B+a₂(v₂)_B+ ....a_m(v_m)_B+...a_k(v_k)_B=0

we have by (1)

   a₁=a₂=...a_m=a_m+1=....=a_k=0

   a₁=a₂=...a_m=0

so

{(v₁)_B,(v₂)_B,....(v_k)_B} is L.I in R^m

conversly suppose that

{(v₁)_B,(v₂)_B....(v_k)_B} is L.I in R^m

let a₁(v₁)_B+a₂(v₂)_B+...a_m(v_m)_B+...a_k(v_k)_B=0

then a₁=a₂=...a_m=...a_k=0 =>(1)

consider

a₁v₁+a₂v₂+...+a_kv_k=0

by (1)

a₁=a₂=....=a_k=0

so

{v_1,v₂...v_k} is L.I in Rⁿ