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 $W$.

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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.