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.