## Exercises

```{exercise-start}

:label: ex1

:label: op_ex1

```

Show that, for any linear subspace $S \subset \mathbb R^n$, $S \cap S^{\perp} = \{0\}$.

```{exercise-end}

```

```{solution-start} ex1

```{solution-start} op_ex1

:class: dropdown

```

If $x \in S$ and $x \in S^\perp$, then we have in particular

```

```{exercise-start}

:label: ex2

:label: op_ex2

```

Let $P = X (X' X)^{-1} X'$ and let $M = I - P$. Show that

$P$ and $M$ are both idempotent and symmetric. Can you give any

```{exercise-end}

```

```{solution-start} ex2

```{solution-start} op_ex2

:class: dropdown

```

```{exercise-start}

:label: ex3

:label: op_ex3

```

Using Gram-Schmidt orthogonalization, produce a linear projection of $y$ onto the column space of $X$ and verify this using the projection matrix $P := X (X' X)^{-1} X'$ and also using QR decomposition, where:

```

```{solution-start} ex3

```{solution-start} op_ex3

:class: dropdown

```