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(section:uge8S-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: The expansion principle

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Let $\mathbf v_1,\dots,\mathbf v_n$ be vectors in $\mathbb{F}^n$ and let $V$ be defined as $V=\mathrm{Span}(\mathbf v_1,\dots,\mathbf v_n)$. Let $d=\dim(V)$, and assume that $\mathbf w_1,\dots,\mathbf w_e$ are linearly independent vectors in $V$, where $e$ is an integer satisfying $e \le d$.

Show that vectors $\mathbf w_{e+1},\dots,\mathbf w_d$ exist in $V$ such that $(\mathbf w_1,\dots,\mathbf w_d)$
is an ordered basis for $V$. In other words, each set of linearly independent vectors in $V$ can be expanded to form an ordered basis for $V$.

```{hint}
:class: dropdown
The vectors $\mathbf w_1,\dots,\mathbf w_e,\mathbf v_1,\dots,\mathbf v_n$ span $V$. Can Theorem 9.2.1 be of use?
```

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## Extra exercise 2: Coset 

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Let $\mathbf v_1,\dots,\mathbf v_n$ be vectors in $\mathbb{F}^n$ and let $V$ be defined as $V=\mathrm{Span}$ $\mathbf v_1,\dots,\mathbf v_n$. A *coset* of $V$ by a vector $\mathbf w \in \mathbb{F}^n$ is defined as follows:

$$\mathbf w + V =\{\mathbf w+\mathbf v \, | \, \mathbf v \in V \}.$$

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### Question a

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Show that the solution set to an inhomogeneous system of linear equations is a coset of the solution set to the corresponding homogeneous system. 

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### Question b

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Show that $\mathbf w_1 + V = \mathbf w_2 + V$ if and only if $\mathbf w_1-\mathbf w_2 \in V$. Conclude that the difference between two particular solutions to an inhomogeneous equation system is a solution to the corresponding homogeneous equation system.