- #1

mathmari

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Let $V,W$ be $\mathbb{R}$-vector spaces and let $\phi:V\rightarrow W$ be a linear map. Let $1\leq k\in \mathbb{N}$ and let $v_1, \ldots , v_k\in V$.

If $V=\text{Lin}(v_1, \ldots , v_k)$ then does it follow that $W=\text{Lin}(\phi (v_1), \ldots , \phi (v_k))$ ?I have shown that $v_1, \ldots , v_k$ are linearly independent iff $\phi (v_1), \ldots , \phi (v_k)$ are linearly independent.

We have that $\phi (V)\subseteq W$ and since $\phi$ is linear we get that $\text{Lin}(\phi (v_1), \ldots , \phi (v_k))\subseteq W$, right?

The dimension of $V=\text{Lin}(v_1, \ldots , v_k)$ is at most $k$, then the dimension of $\text{Lin}(\phi (v_1), \ldots , \phi (v_k))$ is also at most $k$.

But the dimesion of $W$ can be greater. Is that correct?

So we get $\subseteq$ and not necessarily $=$, or not?

What if $W=\text{Lin}(\phi (v_1), \ldots , \phi (v_k))$ then does it follow that $V=\text{Lin}(v_1, \ldots , v_k)$ ?

:unsure: