Question

Let the set of vectors {v₁, ...v_k, ... ,v_n} are basis for subspace V in Rⁿ.

Are the vectors v₁ , .... , v_k are linearly independent too?

Answer 1

S be a subset of V

Let S be a set of v1,v2 ,.....vk,.....,vn vectors

i.e. S={v1,v2,.....vk......vn}

Then set S called basis of V if:

(1) S is linear independent.

(2) S spans vector space V.

So we say S is linear independent because given that S={v1,v2,....vk,...vn} is basis of vector space V.

Also {v1,v2,....,vk} is subset of set S.

And we know that"A subset of linear independent set is also linear independent.

So set {v1,v2,......,vk} is also linear independent .