- #1

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Let $1\leq m, n\in \mathbb{N}$, let $\phi :\mathbb{R}^n\rightarrow \mathbb{R}^m$ a linear map and let $U\leq_{\mathbb{R}}\mathbb{R}^n$, $W\leq_{\mathbb{R}}\mathbb{R}^m$ be subspaces.

I want to show that:

- $\phi (U)$ is subspace of $\mathbb{R}^m$.
- $\phi^{-1} (W)$ is subspace of $\mathbb{R}^n$.

- We have that $\phi (U)=\{\phi (u) \mid u\in U\}$.

Since $U$ is a subspace we have that $0\in U$. Therefore $\phi (0)\in \phi (U)$. Since $\phi$ is a linear map we have that $\phi (0)=0$ and so we get that $0\in \phi (U)$.

Let $\phi (u_1), \phi (u_2)\in \phi (U)$. Then we have that $\phi (u_1)+\phi (u_2)=\phi (u_1+u_2)$, since $\phi$ is linear.

Since $U$ is a subspace we have that since $u_1, u_2\in U$ then $u_1+u_2\in U$. Therefore we get that $\phi (u_1+u_2)\in \phi (U)$ and so we have that $\phi (u_1)+\phi (u_2)\in \phi (U)$.

Let $\lambda\in \mathbb{R}$ and $\phi (u_1)\in \phi (U)$. Then we have that $\lambda \phi (u_1)=\phi (\lambda u_1)$, since $\phi$ is linear.

Since $U$ is a subspace we have that since $\lambda\in \mathbb{R}$ and $u_1\in U$ then $\lambda u_1\in U$. Therefore we get that $\phi (\lambda u_1)\in \phi (U)$ and so we have that $\lambda \phi (u_1)\in \phi (U)$.

That means that $\phi (U)$ is subspace of $\mathbb{R}^m$.

Is everything correct? (Wondering) - We have that $\phi$ is linear. Does it follow then that $\phi^{-1}$ is also linear? (Wondering)