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(section:uge10S-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: Isomorphic vector spaces 

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Let $V$ and $W$ be vector spaces over a field $\mathbb F$. $V$ and $W$ are called *isomorphic* if an invertible linear map $L: V \to W$ exists.

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### Question a

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Let $V$ be a finite-dimensional vector space, and let $n=\dim_{\mathbb F}(V)$. Show that $V$ and $\mathbb F^n$ are isomorphic vector spaces.

```{hint}
:class: dropdown
With an ordered basis $\beta$ for $V$, any vector $\mathbf v \in V$ can be associated with a $\beta$-coordinate vector $[\mathbf v]_{\beta} \in \mathbb F^n$. 
```

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### Question b

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Let both $V$ and $W$ be finite-dimensional vector spaces. Show that $V$ and $W$ are isomorphic if $\dim_{\mathbb F}(V)=\dim_{\mathbb F}(W)$.

In fact, $V$ and $W$ are isomorphic if and only if $\dim_{\mathbb F}(V)=\dim_{\mathbb F}(W)$.

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## Extra exercise 2: The sum of subspaces

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Let $V$ be an $n$-dimensional vector space over a field $\mathbb{F}$, and let $W_1$ and $W_2$ be two subspaces of $V$. The *sum* $W_1+W_2$ of the two subspaces is defined as

$$W_1+W_2=\{\mathbf w_1+\mathbf w_2 \, | \, \mathbf w_1 \in W_1, \mathbf w_2 \in W_2\}.$$

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### Question a

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Show that $W_1+W_2$ is also a subspace of $V$. 

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### Question b

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Show that if $W_1 \cap W_2= \{ \mathbf 0\}$, then every vector $\mathbf v$ in $W_1+W_2$ can be written uniquely in the form $\mathbf v = \mathbf w_1+\mathbf w_2$ for some $\mathbf w_1 \in W_1, \mathbf w_2 \in W_2$. 

```{hint}
:class: dropdown
If $\mathbf w_1+\mathbf w_2=\tilde{\mathbf w}_1+\tilde{\mathbf w}_2$ for $\mathbf w_1, \tilde{\mathbf w}_1 \in W_1$ and $\mathbf w_2, \tilde{\mathbf w}_2 \in W_2$, then $\mathbf w_1- \tilde{\mathbf w}_1 = -\mathbf w_2+ \tilde{\mathbf w}_2$. 
```

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### Question c

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Show that if $W_1 \cap W_2= \{ \mathbf 0\}$, then $V=W_1+W_2$ if and only if $\dim_{\mathbb F}(W_1)+\dim_{\mathbb F}(W_2)=\dim_{\mathbb F}(V)$.