# Thread: Matrix Representation of Linear Transformations (2)

1. ## Matrix Representation of Linear Transformations (2)

Let V and W be vector spaces, and let S be a subset of V. Define S^0 = {T Є L (V,W): T(x) = 0 for all x Є 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 is contained in S2, then S2^0 is contained in S1^0

Thanks!!

2. Let V and W be vector spaces, and let S be a subset of V. Define S^0 = {T Є L (V,W): T(x) = 0 for all x Є S}. Prove the following statements:

a) S^0 is a subspace of L(V,W)
To show $\displaystyle S^0$ is a subspace you need to prove $\displaystyle T_1,T_2 \in S^0$ then $\displaystyle T_1 + T_2\in S^0$. Next you need to prove that if $\displaystyle T \in S^0$ and $\displaystyle k \in K$ (the field you are working over) then $\displaystyle kT \in S^0$

b) If S1 and S2 are subsets of V and S1 is contained in S2, then S2^0 is contained in S1^0
To show $\displaystyle S_2^0 \subseteq S_1^0$ you need to show if $\displaystyle T \in S_2^0$ then $\displaystyle T \in S_1^0$.
This is straightforward.