# Linear Algebra help

• Jul 13th 2008, 10:08 AM
JCIR
Linear Algebra help
Let V and W be vectors spaces, and let S be subset of V. Define S^0 = { T \in \L(v,w) : T(x) =0 for all x \in \ S.

Prove the following statements
a) S^0 is a subspace of L(v,w)
b)If S1 and S2 are subsets of V and S1 subset S2, then S^0_2 less than or equal S^0_1.
• Jul 13th 2008, 10:15 AM
ThePerfectHacker
Quote:

Originally Posted by JCIR
Let V and W be vectors spaces, and let S be subset of V. Define S^0 = { T \in \L(v,w) : T(x) =0 for all x \in \ S.

Prove the following statements
a) S^0 is a subspace of L(v,w)
b)If S1 and S2 are subsets of V and S1 subset S2, then S^0_2 less than or equal S^0_1.

What do you need to show that $S^0$ is a subspace?
You need to show it is closed under vector addition and scalar multiplication.
Can you show that?
• Jul 13th 2008, 10:18 AM
JCIR
Quote:

Originally Posted by ThePerfectHacker
What do you need to show that $S^0$ is a subspace?
You need to show it is closed under vector addition and scalar multiplication.
Can you show that?

It will be helpful if you can show condition 2. I will be able to show that 0 is in there and the it is closed under scalar multiplication.

Mainly part b is the one I dont get.
• Jul 13th 2008, 10:26 AM
ThePerfectHacker
Quote:

Originally Posted by JCIR
Let V and W be vectors spaces, and let S be subset of V. Define S^0 = { T \in \L(v,w) : T(x) =0 for all x \in \ S.

b)If S1 and S2 are subsets of V and S1 subset S2, then S^0_2 less than or equal S^0_1.

You want to prove $S^0_2 \subseteq S_0^1$. The way we show this is by showing if $T\in S_2^0$ then $T \in S_1^0$. By definition $T \in S_2^0$ means $T(\bold{x}) = \bold{0}$ for all $\bold{x}\in S_2$. But $S_1\subseteq S_2$ so $T(\bold{x}) = \bold{0}$ for all $\bold{x} \in S_1$. Thus, $T\in S_1^0$.