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mathmari

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MHB

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Hey! :giggle:

Let $C$ be a $\mathbb{R}$-vector space, $1\leq n\in \mathbb{N}$ and let $\phi_1, \ldots , \phi_n:V\rightarrow V$ be linear maps.

I have shown by induction that $\phi_1\circ \ldots \circ \phi_n$ is then also a linear map.

I want to show now by induction that if $V$ is finite then $\text{Rank}(\phi_1\circ \ldots \circ \phi_n)\leq \min \{\text{Rank}(\phi_i)\mid 1\leq i\leq n\}$.

Does it hold in general that $\text{Im}(f\circ g)\subseteq \text{Im}(f)$ and $\text{Im}(f\circ g)\subseteq \text{Im}(g)$ and so we get $\text{Rank}(f\circ g)\leq \text{Rank}(f)$ and $\text{Rank}(f\circ g)\leq \text{Rank}(g)$ ?

:unsure:

**Question 1:**Let $C$ be a $\mathbb{R}$-vector space, $1\leq n\in \mathbb{N}$ and let $\phi_1, \ldots , \phi_n:V\rightarrow V$ be linear maps.

I have shown by induction that $\phi_1\circ \ldots \circ \phi_n$ is then also a linear map.

I want to show now by induction that if $V$ is finite then $\text{Rank}(\phi_1\circ \ldots \circ \phi_n)\leq \min \{\text{Rank}(\phi_i)\mid 1\leq i\leq n\}$.

__Base Case__: For $n=1$ we have that $\text{Rank}(\phi_1)\leq \min \{\text{Rang}(\phi_1)\}$, so the equality holds.__Inductive Hypothesis__: We suppose that it holds for $n=m$, so $\text{Rank}(\phi_1\circ \ldots \circ \phi_m)\leq \min \{\text{Rank}(\phi_i)\mid 1\leq i\leq m\}$. (IV)__Inductive Step__: We want to show that it holds for $n=m+1$, i.e. that $\text{Rank}(\phi_1\circ \ldots \circ \phi_m\circ \phi_{m+1})\leq \min \{\text{Rank}(\phi_i)\mid 1\leq i\leq m+1\}$.Does it hold in general that $\text{Im}(f\circ g)\subseteq \text{Im}(f)$ and $\text{Im}(f\circ g)\subseteq \text{Im}(g)$ and so we get $\text{Rank}(f\circ g)\leq \text{Rank}(f)$ and $\text{Rank}(f\circ g)\leq \text{Rank}(g)$ ?

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