Let V and W be vector spaces over R. Show that T (0v ) = 0w,
where T is any linear mapping from V to W , and 0v and 0w are the
additive identities of V and W respectively.


We are tasked with showing that if \( T \) is a linear map from the vector space \( V \) to the vector space \( W \), then \( T(0_v) = 0_w \), where \( 0_v \) and \( 0_w \) are the additive identities of \( V \) and \( W \), respectively.

### Proof:

1. **Linear map definition:**
   By the definition of a linear map \( T: V \to W \), the map satisfies two properties for all vectors \( u, v \in V \) and scalar \( \alpha \in \mathbb{R} \):
   - **Additivity**: \( T(u + v) = T(u) + T(v) \)
   - **Homogeneity**: \( T(\alpha v) = \alpha T(v) \)

2. **Start with the additive identity in \( V \):**
   Let \( 0_v \) be the additive identity in \( V \). By definition, for any vector \( v \in V \),
   \[
   v + 0_v = v.
   \]
   
3. **Apply the linear map \( T \):**
   Using the additivity property of the linear map, we apply \( T \) to both sides of the equation:
   \[
   T(v + 0_v) = T(v).
   \]
   By additivity of \( T \), we have:
   \[
   T(v + 0_v) = T(v) + T(0_v).
   \]
   Thus, we get the equation:
   \[
   T(v) + T(0_v) = T(v).
   \]

4. **Solve for \( T(0_v) \):**
   Subtract \( T(v) \) from both sides:
   \[
   T(0_v) = 0_w,
   \]
   where \( 0_w \) is the additive identity in \( W \).

Thus, we have shown that \( T(0_v) = 0_w \).

### Conclusion:
This completes the proof that the linear map \( T \) sends the additive identity of \( V \) to the additive identity of \( W \).

---

Would you like further clarification on any step of the proof? Or do you have any additional questions?

Here are some related questions to expand on this topic:
1. What is the kernel of a linear map, and how is it related to the additive identity?
2. How can we prove that a linear map sends scalar multiples of the additive identity to zero?
3. Can this result be generalized to vector spaces over other fields, not just \( \mathbb{R} \)?
4. What does it mean for a linear map to be injective or surjective, and how is the additive identity involved?
5. How do we show that the image of a linear map always contains the zero vector?

**Tip:** In any proof involving linear maps, always start by using the properties of additivity and homogeneity, as they are the backbone of linearity and often simplify the problem significantly.

Let V and W be vector spaces over R. Show that T (0v ) = 0w, where T is any linear mapping from V to W, and 0v and 0w are the additive identities of V and W respectively.

Explore the proof that any linear map T from a vector space V to a vector space W preserves the zero vector, meaning T(0v) = 0w. This step-by-step solution covers key concepts in linear algebra, including vector space properties and linear transformations. Suitable for undergraduate students studying linear algebra.

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