1. ## Proof

Show that if U is a subspace of a vector space V that is in turn a subspace of a vector space W, then U is a subspace of W

2. Originally Posted by skittle
Show that if U is a subspace of a vector space V that is in turn a subspace of a vector space W, then U is a subspace of W
$\displaystyle U\subset V\subset W$

By transitivity, $\displaystyle U\subset W$.

I think you can just do this.

3. Just say everything in V is in W, and everything in U is in V.

Therefore everything in U must be in W.

4. What dwsmith and Prove It suggest shows that U is a subset of W. You still need to show it is a vector space by showing that it is closed under addition and scalar multiplication.

5. Originally Posted by HallsofIvy
What dwsmith and Prove It suggest shows that U is a subset of W. You still need to show it is a vector space by showing that it is closed under addition and scalar multiplication.
If U is a subset of W, and it is a vector space over the same field with addition and scalar multiplication identical with the restriction of the equivalent operations on W to U then it is a subspace of W. That U is a subspace of V a subspace of W is sufficient to justify this last requirement.

CB