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(section:uge9S-extra)=
# Extra (optional) exercises -- Long Day

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These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

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## Extra exercise 1: Considering complex vector spaces as real vector spaces 

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Let $V$ be a finite-dimensional complex vector space. We will write $n=\dim_{\mathbb C}(V)$.

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### Question a

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Confirm that $V$ can be considered a real vector space since addition of vectors and the scalar-vector product $c \cdot \mathbf v$ for $c \in \mathbb{R}$ and $\mathbf v \in V$ are defined by using the fact that $V$ is a complex vector space.  

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### Question b

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Show that $\dim_{\mathbb R}(V)=2\dim_{\mathbb C}(V)=2n$. 

```{hint}
:class: dropdown
Let $(\mathbf v_1,\dots,\mathbf v_n)$ be an ordered basis for $V$ considered as a complex vector space. Can you find an ordered basis for $V$ considered as a real vector space? 
```

```{hint}
:class: dropdown
Check that $(\mathbf v_1,i\cdot \mathbf v_1,\dots,\mathbf v_n,i\cdot \mathbf v_n)$ is an ordered basis for $V$ considered as a real vector space. 
``` 

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## Extra exercise 2: Complete flags

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Let $n$ and $m$ be natural numbers, $V$ an $n$-dimensional vector space over a field $\mathbb{F}$ and $W$ an $m$-dimensional subspace of $V$.

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### Question a

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Assume that $m <n-1$. Show that a subspace $\tilde{W}$ of $V$ exists such that $W$ is a subspace of $\tilde{W}$. 

```{hint}
:class: dropdown
Assume that $(\mathbf w_1,\dots,\mathbf w_m)$ is an ordered basis for $W$. Use your answers to Extra exercise 1 from Long Day in Week 8.
```

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### Question b

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A sequence of subspaces $W_0,W_1,\dots,W_n$ of $V$ that satisfy $\dim(W_i)=i$ for all $i$, as well as $W_0 \subset W_1 \subset \cdots \subset W_n$, is called a *complete flag* in $V$. Show that a complete flag exists in $V$ such that $W_m=W$.

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### Question c

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Now let $V$ be the real vector space $\mathbb{R}^4$. Construct two complete flags $V_0,V_1,V_2,V_3,V_4$ and $W_0,W_1,W_2,W_3,W_4$ in $\mathbb{R}^4$ such that $\dim(V_2 \cap W_2)=0$ and $\dim(V_3 \cap W_3)=2$. Is it possible to construct two complete flags in $\mathbb{R}^4$ that satisfy $\dim(V_2 \cap W_2)=0$ and $\dim(V_3 \cap W_3)=3$?