Note that you can use the same c1, c2, and c3 for both parts.

[RedXIII]

Homework Statement

1) Is S a subspace of R^n?

1.1) Given n=4 and a vector is in S if it is in the span of e1, e2 or in the span of e3, e4 where e1...e4 is the canonical basis of R^4

1.2) Given n=3 and S is a sphere of radius 1.

2) Let S be a subspace of R^10 with basis v1; v2; v3. Show that the vectors 2v1 + 4v2; v2 + 2v3; 5v3 also form a basis of S.

The Attempt at a Solution

1.1) No, because if the vector only spans e1, e2 or e3, e4, then it would only generate a plane in R^4?

1.2) I was thinking yes because a sphere is a 3-d object, thus would be a subspace of itself in R^3. However, if the radius is one then would that be false because if I were to add the vectors of the sphere, it would be outside the sphere?

2) From what I read, I think I would have to prove that they are linear independent then prove that they can generate R^10 (both of which are already proven if v1,v2,v3 is the basis of R^10) but how would I show it without any actual numbers?

Thanks!

 

[RedXIII]

Anybody?

 

[vela]

What's the definition of a subspace? For problem 1, you want to show that those cases either satisfy or don't satisfy the definition, so if you don't know the definition, that's the first thing you need to look up.

For problem 2, you need to be bit more careful. The problem doesn't say v₁, v₂, and v₃ form a basis for R¹⁰. It says they form a basis for S, a subspace of R¹⁰. Now you're given three vectors — let's call them w₁, w₂, and w₃ — and you want to show they form a basis for S. You need to show two things:

1. w₁, w₂, and w₃ span S. That is, if x is in S, then you can find constants c₁, c₂, and c₃ such that x=c₁w₁+c₂w₂+c₃w₃.
2. w₁, w₂, and w₃ are linearly independent. That is, if c₁w₁+c₂w₂+c₃w₃=0, then c₁=c₂=c₃=0 is the only solution.