Is W a Subspace of V^3 Given These Vectors?

[fishingspree2]

We have W = [v₁,v₂,v₃,v₄]

v₁=i-k
v₂=i+j+k
v₃=j+2k
v₄=2i+j

Show that W is a subspace of V³.
first, vector 0 is obviously in W.

then,
let u = n₁v₁+n₂v₂+n₃v₃+n₄v₄ ∈ W
and v = s₁v₁+s₂v₂+s₃v₃+s₄v₄ ∈ W

and p ∈ reals

then u+pv
=(n₁+ps₁)v₁+(n₂+ps₂)v₂+(n₃+ps₃)v₃+(n₄+ps₄)v₄
∈ W

Am i completely proving that W is a subspace of V³ (the 3 dimensional space)? I am not quite sure because I am not even using the explicit given vectors.
Thank you very much

 

[Anonymous217]

By V^3, do you mean R^3? It seems fine since you proved it in the general case by satisfying the three conditions.

 

[arkajad]

I would suggest calculating:

v3-v4+2v1
v3+v4-2v2

and what these results tell you?

 

[HallsofIvy]

Your post is very confusing! You titled this "show that W is a subset of V^3" which is quite different from saying "show that W is a subspace of V^3".

You then say W= [v1, v2, v3, v4]. Are we to assume that you mean that W is the span of those vectors? And are we to assume those vectors are in V^3?

The span of a collection of vectors is always a subspace of the space they exist in, pretty much from the definition of "span"- since every linear combination is, by definition, in the "span", certainly sums and scalar products are.

 

[arkajad]

Probably the problem was to show that it is a proper subspace.