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(section:uge6L-extra)=
# Extra (optional) exercises -- Short 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: Uniqueness of the reduced row-echelon form

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Section 6.5 in the textbook (which is not part of the syllabus) explains that a given matrix has a uniquely determined reduced row-echelon form. The goal of this extra exercise is to understand the proof of Theorem 6.5.1.

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### Question a

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Read Theorem 6.5.1, but not its proof. How would you go about proving this theorem?

Now read the first paragraph of the proof:

> From Theorem 6.2.1, we know that the homogeneous systems of linear equations with coefficient matrices $\mathbf A$, $\mathbf B_1$, and $\mathbf B_2$ all have exactly the same solutions. The idea of the proof is to show that the homogeneous systems of linear equations with coefficient matrices $\mathbf B_1$ and $\mathbf B_2$ only can have the same solutions if $\mathbf B_1 = \mathbf B_2$. Moreover, we use induction on $n$, the number of columns.

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### Question b

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The first step in the proof is to show that the base case of the induction holds. Try proving Theorem 6.5.1 for $n=1$.

```{hint}
:class: dropdown
If you get stuck, then just look at the next paragraph in the proof of Theorem 6.5.1 in the textbook.
```

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### Question c

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The next step in the proof of Theorem 6.5.1 is the induction step. First, try yourself to carry out the proof. If you get stuck, then look in the textbook to see how the induction step is done there.