Suppose 𝑣1
, … , 𝑣𝑚 is a list of vectors in 𝑉. Define 𝑇 ∈ ℒ(𝐅
𝑚, 𝑉) by
𝑇(𝑧1
, …, 𝑧𝑚) = 𝑧1𝑣1 + ⋯ + 𝑧𝑚𝑣𝑚.
(a) What property of 𝑇 corresponds to 𝑣1
, … , 𝑣𝑚 spanning 𝑉?

The linear map \( T \in \mathcal{L}(\mathbb{F}^m, V) \) is defined by

\[
T(z_1, \dots, z_m) = z_1v_1 + \dots + z_mv_m
\]

where \( z_1, \dots, z_m \in \mathbb{F} \) are scalars, and \( v_1, \dots, v_m \in V \) are vectors in a vector space \( V \).

### (a) What property of \( T \) corresponds to \( v_1, \dots, v_m \) spanning \( V \)?

The property of \( T \) that corresponds to the vectors \( v_1, \dots, v_m \) spanning \( V \) is that **\( T \) is surjective** (or onto).

#### Explanation:
- A set of vectors \( v_1, \dots, v_m \) spans the vector space \( V \) if every vector in \( V \) can be written as a linear combination of \( v_1, \dots, v_m \). 
- This means for every \( v \in V \), there exists a tuple \( (z_1, \dots, z_m) \in \mathbb{F}^m \) such that:
  \[
  v = z_1v_1 + \dots + z_mv_m = T(z_1, \dots, z_m)
  \]
- Hence, the condition that \( v_1, \dots, v_m \) span \( V \) is equivalent to saying that \( T \) is surjective. In other words, the image of \( T \) covers the entire space \( V \), meaning \( \text{Im}(T) = V \).

Would you like further details on the concept of surjectivity or how it's proven in this context?

Here are 5 related questions to think about:
1. What does it mean for \( T \) to be injective in this context?
2. How does the dimension of \( \mathbb{F}^m \) relate to the dimension of \( V \) when \( T \) is surjective?
3. What happens if \( v_1, \dots, v_m \) are linearly dependent?
4. Can the vectors \( v_1, \dots, v_m \) span \( V \) if \( m < \dim(V) \)?
5. How would you describe the kernel of \( T \) when \( v_1, \dots, v_m \) span \( V \)?

**Tip**: Surjectivity means that for every element in the codomain, there is a preimage in the domain. This is crucial for spanning sets, as they need to cover the entire space.

Suppose v1, … , vm is a list of vectors in V. Define T ∈ ℒ(Fm, V) by T(z1, …, zm) = z1v1 + ⋯ + zmvm. (a) What property of T corresponds to v1, … , vm spanning V?

Understand the relationship between a set of vectors spanning a vector space and the surjectivity of a corresponding linear map. Explore how a linear map T(z1, ..., zm) = z1v1 + ... + zmvm connects with spanning vectors in V. Suitable for undergraduate students studying linear algebra.

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