Linear transformations - 2 opposite claims, solution attempt included

Question

Suppose $V, W$ are vector spaces and $ T: V \to W $ is a linear transformation.

$v_1, v_2, ... , v_k \in V$.

Prove or disprove:

• If $span( v_1, v_2, ... , v_k) = V$, then $span(T(v_1), ... , T(v_k) = W$.

• If $span(T(v_1), ... , T(v_k)) = W$, then $span( v_1, v_2, ... , v_k) = V$.

My solution goes:

• The first claim is false. let $, V, W = \mathbb R$, and $ T: \mathbb R \to \mathbb R$. We define $T$ as $T(v) = 0$, and $v_1 = [1]$. Therefore, $span(v_1) =\mathbb R$, but $span(T(v_1)) = span(0) \neq \mathbb R$.

• Second claim: We can easily prove that if $(T(v_1),...,T(v_n))$ is linearly independent, then $(v_1,...,v_n)$ is linearly independent as well. Is this enough to conclude that since we know $span(T(v_1), ... , T(v_k)) = W$, then $T(v_1), ... , T(v_k)$ is linearly independent, therefore $ v_1, v_2, ... , v_k$ is linearly independent, and thus $span( v_1, v_2, ... , v_k) = V$?

Thanks in advance!

Answer 1

The first you did great. For the second consider $W=\{0\}$