# Notes on Euclidean Spaces

## Real euclidean spaces⌗

Real euclidean spaces have definitions of inner product and norm. Examples in \(\mathbb R^n\):

- The usual inner product
- The unit-radius circumference when considering an unusual inner product
- Cauchy-Schwarz inequality

Let \(V\) be a real vector space. A form or real function \[
\begin{aligned} \langle\cdot,\cdot\rangle\colon V\times V &\rightarrow \mathbb R
\\ (x, y) &\mapsto \langle x,y\rangle \end{aligned}\ \] is said to be an
*inner product* if, for all \(x, y, z \in V\) and all \(\alpha \in \mathbb
R\),

- \(\langle x,y \rangle = \langle y, x \rangle\)
- \(\langle \alpha x, y\rangle = \alpha\langle x, y\rangle\)
- \(\langle x+y, z\rangle = \langle x, z\rangle + \langle y, z\rangle\)
- \(\langle x, x\rangle \geq 0 \wedge (\langle x, x\rangle = 0 \implies x = 0)\)

A real linear space \(V\) equipped with an inner product is called an (real)
**Euclidean Space**.

### Examples⌗

- Usual inner product in \(\mathbb R^n\)
- \(\mathbb R^2\) \[ \begin{aligned} \langle x, y\rangle &= \lVert x\rVert\lVert y\rVert \cos\theta \\ &= x_1 y_1 + x_2 y_2 \quad\text{in }\mathbb R^2 \end{aligned}\ \] where \(\theta\in[0, \pi]\) is the angle between the vectors \(x\) and \(y\). Note that the norm of the vector \(x\) satisfies \[ \lVert x\rVert^2 = \langle x, x\rangle \]
- \(\mathbb R^n\) \[ \begin{aligned} \langle x, y\rangle &= x_1 y_1 + x_2 y_2 + \cdots + x_n y_n \\ \langle x, y\rangle &= y^Tx = x^Ty \end{aligned}\ \]

- Another inner product in \(\mathbb R^2\)
**Exercise.**Determine the circumference \(C\) of radius \(1\) and centered at \((0, 0)\) \[ C = {(x_1, x_2)\in\mathbb R^2\colon \lVert(x_1, x_2)\rVert = 1} \] considering- The usual inner product
- The inner product \[\langle (x_1, x_2), (y_1, y_2) \rangle = \frac{1}{9} x_1y_1 + \frac{1}{4} x_2y_2 \]

### Norm and the triangle inequality⌗

For all vectors \(x\in V\), we define the **norm** of \(x\) as \[ \lVert
x\rVert = \sqrt{\langle x, x\rangle} \] such that, for all \(x\in V\) and all
\(\alpha\in\mathbb R\) we have

- \(\lvert x\rVert\geq 0\qquad \text{and}\qquad \lVert x\rVert \iff x = 0\)
- \(\lVert \alpha x\rVert = \lvert\alpha\rvert\lVert x\rVert\)
- \(\lVert x + y\rVert\leq\lVert x\rVert + \lVert y\rVert \qquad\qquad \text{(triangle inequality)} \)

A function \(V \to \mathbb R\) that satisfies the above conditions is said to
be a **norm** defined in \(V\).

The proof that the function we defined earlier and called “inner product” satisfies the triangle inequality will be done at a later point, since it relies on the Cauchy-Schwarz inequality.

### Cauchy-Schwarz Inequality⌗

**Theorem 1.** *Let \(V\) be an euclidean space. For all \(x, y \in V\) we
have* \[ \lvert\langle x, y\rangle\rvert \leq \lVert x\rVert \lVert y\rVert \]
Note that in \(\mathbb R^2\) and \(\mathbb R^3\) we have: \[ \langle x,
y\rangle = \lVert x\rVert\lVert y\rVert\cos\theta \] where \[ \lvert\langle x,
y\rangle\rvert = \lVert x\rVert\lVert y\rVert\lvert \cos\theta\rvert \leq \lVert
x\rVert \lVert y\rVert \]

### Distance⌗

For all \(x, y \in V\), we define the **distance from \(x\) to \(y\)** as
\[ d(x, y) = \lVert x - y\rVert \]

### Parallelogram Law⌗

For all vectors \(x, y \in V\), we have \[ \lVert x + y\rVert^2 + \lVert x - y\rVert^2 = 2(\lVert x\rVert^2 + \lVert y\rVert^2) \]

### Example⌗

An inner product in \(\mathbb M_{2\times 2}(\mathbb R)\).

For all matrices \(A, B \in \mathbb M_{2\times 2}(\mathbb R)\) we define \[
\begin{aligned} \langle A, B\rangle &= tr(B^T A) \\ &= \sum^2_{i,
j=1}{a_{ij}b_{ij}} \end{aligned}\] with \(A = [a_{ij}]\) and \(B =
[b_{ij}]\)^{1}. Note that, letting \(B_c\) be the canonical basis of
\(\mathbb M_{2\times 2}(\mathbb R)\)

\[\langle A, B \rangle_{\mathbb M_{2\times 2}(\mathbb R)} = \langle (A)_{B_c}, (B)_{B_c}\rangle_{\mathbb R^4}\]

Meaning that the inner product defined above respects the isomorphism \( A \mapsto (A)_{B_c}\) between \( \mathbb{M}_{2 \times 2}(\mathbb R) \) and \(\mathbb R^4\)

### Proof of the triangle inequality⌗

\[ \begin{aligned} \lVert x + y\rVert^2 &= \langle x+y, x+y\rangle \\ &= \langle x, x\rangle + 2\langle x, y\rangle + \langle y, y\rangle \\ &= \lVert x\rVert^2 + 2\langle x, y\rangle + \lVert y\rVert^2 \qquad (\text{Inner product in terms of the norm})\\ &\leq \lVert x\rVert^2 + 2\lvert\langle x, y\rangle\rvert + \lVert y\rVert^2 \\ &\leq \lVert x\rVert^2 + 2\lVert x\rVert\lVert y\rVert + \lVert y\rVert^2 \qquad (\text{Cauchy-Schwartz inequality}) \\ &=(\lVert x\rVert + \lVert y\rVert)^2 \end{aligned}\]

Where \[ \lVert x + y\rVert \leq \lVert x\rVert + \lVert y\rVert \qquad_\blacksquare\]

## Gram Matrix⌗

Let \(V\) be a real euclidean space, and \(B = (b_1, b_2, \ldots, b_n)\) a basis of \(V\). With \(x, y \in V\) such that \(x_B = (\alpha_1, \alpha_2, \ldots, \alpha_n)\) and \(y_B = \beta_1, \beta_2, \ldots, \beta_n\), we have \[\begin{aligned} \langle x, y\rangle &= \langle\alpha_1 b_1 + \alpha_2 b_2 + \cdots + \alpha_n b_n, \beta_1 b_1 + \beta_2 b_2 + \cdots + \beta_n b_n\rangle \\ &= \begin{bmatrix}\beta_1 & \beta_2 & \ldots & \beta_n \end{bmatrix} \underbrace{\begin{bmatrix}\langle b_1, b_1\rangle & \langle b_2, b_1\rangle &\ldots & \langle b_n, b_1\rangle \\ \langle b_1, b_2\rangle & \langle b_2, b_2\rangle &\ldots & \langle b_n, b_2\rangle \\ \vdots \\ \langle b_1, b_n\rangle & \langle b_2, b_n\rangle &\ldots & \langle b_n, b_n\rangle \\ \end{bmatrix}}_G \begin{bmatrix}\alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n\end{bmatrix} \end{aligned}\ \]

Therefore, given an inner product in \(V\) and a basis \(B\), it is possible to determine a matrix \(G\) such that \[\langle x,y\rangle = y_B^T Gx_B\]

This matrix \(G = [g_{ij}]\), where for all \(i, j = 1, \ldots, n\) we have
\(g_{ij} = \langle b_j, b_i \rangle\) is called the **Gram matrix** of the set
of vectors \(\{b_1, b_2, \ldots, b_n\}\).

Note that:

- \(G\) is a symmetric (\(G = G^T\)) \(n\times n\) real matrix.
- For all non-null vectors \(x\in V\) \[x_B^T Gx_b > 0\]

A square real matrix \(A\) of order \(k\) is said to be **positive
definite** if, for all non-null vectors \(x\in\mathbb R^n\), \(x^T Ax >
0\)

**Proposition 1.** *A symmetric real matrix is positive definite iff all your
proper values are positive.*

**Theorem 2.** *Let \(A\) be a symmetric real matrix of order \(n\). The
following statements are equivalent.*

*The expression \[\langle x, y\rangle = y^T Ax\] defines an inner product in \(\mathbb R^n\)**\(A\) is positive definite.*

### Exercise⌗

Consider that \(\mathbb R^n\) is equipped with the canonical basis \(\mathcal{e}_n\). What is the Gram matrix \(G\) that corresponds to the usual inner product in \(\mathbb R^n\)? Also, which Gram matrix corresponds to the inner product in item (2) of the previous exercise?

## Complex euclidean spaces and orthogonal vectors⌗

Example of complex euclidean space: usual inner product in \(\mathbb C^n\).

Let \(V\) be a complex vector space. A complex function or form
\[ \begin{aligned}\langle\cdot,\cdot\rangle\colon V \times V &\to \mathbb C
\\ (x, y) &\mapsto \langle x, y\rangle\end{aligned} \]
is said to be an **inner product** if, for all \(x, y, z \in V\) and all
\(\alpha \in \mathbb C\)

- \(\langle x, y\rangle = \overline{\langle y, x\rangle}\)
- \(\langle \alpha x, y\rangle = \alpha\langle x,y\rangle\)
- \(\langle x + y, z\rangle = \langle x, z\rangle + \langle y, z\rangle\)
- \(\langle x, x\rangle \geq 0 \wedge (\langle x, x\rangle = 0 \implies x = 0)\)

A complex vector space \(V\) equipped with an inner product is called a (complex)
**euclidean space**.

Much like with real euclidean spaces, we define the **norm** of a vector as
\[\lVert x\rVert = \sqrt{\langle x, x\rangle}\] and the **distance from
\(x\) to \(y\) as \[d(x, y) = \lVert x - y\rVert\]**

**Example.** Usual inner product in \(\mathbb C^n\). Let \(x = (x_1, x_2,
\ldots, x_n)\) and \(y_1, y_2, \ldots, y_n\) be vectors in \(\mathbb C^n\),
we define \[\langle x, y\rangle = x_1\overline{y}_1 + x_2\overline{y}_2 +
\cdots + x_n\overline{y}_n\] and therefore \[\langle x, y\rangle =
\overline{y}^T x\] Regarding the norm we have \[\lvert x\rVert^2 =
\langle x, x\rangle = x_1\overline{x}_1 + x_2\overline{x}_2 + \cdots +
x_n\overline{x}_n\] or \[\lVert x\rVert = \sqrt{\lVert x, x\rVert} =
\sqrt{\lvert x_1\rvert^2 + \lvert x_2\rvert^2 + \cdots + \lvert x_n\rvert^2}\]

All the remaining results that were presented regarding real euclidean spaces are also true for complex euclidean spaces (Cauchy-Schwartz, triangle inequality, parallelogram law, …).

### Complex Gram Matrix⌗

Let \(V\) be a complex euclidean space, and let \(B = (b_1, b_2, \ldots, b_n)\) be a basis of \(V\). With \(x, y \in V\) such that \(x_B=(\alpha_1, \alpha_2, \dots, \alpha_n)\) and \(y_B=(\beta_1, \beta_2, \dots, \beta_n)\), we have \[ \begin{aligned} \langle x, y\rangle &= \langle\alpha_1 b_1 + \alpha_2 b_2 + \cdots + \alpha_n b_n, \beta_1 b_1 + \beta_2 b_2 + \cdots + \beta_n b_n\rangle \\ &= \begin{bmatrix}\overline{\beta}_1 & \overline{\beta}_2 & \ldots & \overline{\beta}_n\end{bmatrix} \underbrace{\begin{bmatrix}\langle b_1, b_1\rangle & \langle b_2, b_1\rangle &\ldots & \langle b_n, b_1\rangle \\ \langle b_1, b_2\rangle & \langle b_2, b_2\rangle &\ldots & \langle b_n, b_2\rangle \\ \vdots \\ \langle b_1, b_n\rangle & \langle b_2, b_n\rangle &\ldots & \langle b_n, b_n\rangle \\ \end{bmatrix}}_G \begin{bmatrix}\alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n\end{bmatrix} \end{aligned}\ \]

Therefore, given an inner product in \(V\) and a basis \(B\), it is possible to determine a matrix \(G\) such that \[\langle x,y\rangle = \overline{y}_B^T Gx_B\]

This matrix \(G = [g_{ij}]\), where for all \(i, j = 1, \ldots, n\) we have
\(g_{ij} = \langle b_j, b_i \rangle\) is called the **Gram matrix** of the
set of vectors \(\{b_1, b_2, \ldots, b_n\}\).

Note that:

- \(G\) is an \(n\times n\) complex matrix such that (\(G = \overline{G}^T\)).
- For all non-null vectors \(x\in V\) \[\overline{x}_B^T Gx_b > 0\]

A complex square matrix \(A\) of order \(k\) is said to be **hermitian** if
\(A = \overline{A}^T\). Note that the spectrum \(\sigma(A)\) of a hermitian
is contained in \(\mathbb R\).

A hermitian matrix \(A\) of order \(k\) is said to be **positive definite**
if, for all non-null vectors \(x\in\mathbb C^n\), \(\overline{x}^T Ax > 0\).

**Proposition 2.** *A hermitian matrix is positive definite iff all of it’s
proper values are positive.*

**Theorem 3.** *Let \(A\) be a hermitian matrix of order \(n\). The
following statements are equivalent.*

*The expression \[\langle x, y\rangle = \overline{y}^T Ax\] defines an inner product in \(\mathbb C^n\)**A is positive definite.*

### Angle between two vectors⌗

Let \(x\) and \(y\) be non-null vectors belonging to some **real** euclidean
space \(V\). We define the *angle* between the vectors \(x\) and \(y\) as
being the angle \(\theta\), with \(0\leq\theta\leq\pi\), such that
\[\cos\theta = \frac{\langle x, y\rangle}{\lVert x\rVert \lVert y\rVert}\]
With Cauchy-Schwartz we can see that \(\lvert\cos\theta\rvert\leq 1\).

Let \(x\) and \(y\) be (possibly null) vectors belonging to some **real or
complex** euclidean space \(V\). The vectors \(x\) and \(y\) are said to
be **orthogonal**, written \(x \perp y\), if
\[\langle x, y\rangle = 0\]

**Exercise.** What are the orthogonal vectors to \(v = (1, 1, 0)\) considering
\(\mathbb R^3\) with the usual inner product?

**Theorem 4. (Pythagoras Theorem)** Let \(x\) and \(y\) be orthogonal
vectors of some euclidean space \(V\). Then
\[\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y \rVert^2\]

*Proof.* Exercise

## Orthogonal complement⌗

Let \(X\) be a subspace of an euclidean space \(V\). We say that \(u\) is
**orthogonal to** \(X\) if \(u\) is orthogonal to all elements of \(X\).
We write this \(u \perp W\).

For example, \((1, 1, 0)\) is orthogonal to the plane \(S\) of the previous exercise.

Let \(W\) be a subspace of \(V\). The **orthogonal complement** of \(W\),
written \(W^\perp\), is defined as
\[W^\perp = \{u\in V\colon u\perp W\}\]

**Exercise.** Determine the orthogonal complement of the line generated by the
vector \((1, 1, 0)\).

**Proposition 3.** \(W^\perp\) is a subspace of V.

**Proposition 4.** *Let \(W\) be a linear subspace of an euclidean space
\(V\) and let \(\{u_1, u_2, \ldots, u_k\}\) be a generator set of
\(W\). Then, \(e\in V\) is orthogonal to \(W\) iff it is orthogonal to
\(\{u_1, u_2, \ldots, u_k\}\).*

**Corollary 1.** *In the conditions of the previous proposition, \(u\in V\) is
orthogonal to \(W\) iff it is orthogonal to a basis of \(W\).*

**Exercise.** Determine the orthogonal complement of the plane \(W\in\mathbb
R^3\) with the cartesian equation \(x=y\).

**Solution.** \(W^\perp\) is the line described by the equations
\[\begin{cases}x = -y \\ z = 0\end{cases}\qquad\qquad\textbf{cartesian equations}\]
or
\[(x, y, z) = t(-1, 1, 0)\qquad(t\in\mathbb R)\qquad\qquad\textbf{vector equation}\]
or
\[\begin{cases}x = -t \\ y = t \\ z = 0\end{cases}\qquad(t\in\mathbb
R)\qquad\qquad\textbf{parametric equations}\]

**Proposition 5.** *Let \(W\) be a subspace of an euclidean space \(V\).*

- \(W\cap W^\perp = 0\)
- \(W^{\perp\perp} = W\)

A subset \(X\) of an euclidean space \(V\) is said to be an **orthogonal
set** if, for all \(x, y\in X\) with \(x \neq y\) we have \(x \perp y\).

**Question.** Let \(X\) be an orthogonal set not containing the null vector.

- If \(X\subseteq \mathbb R^2\), how many vectors does \(X\) have at most?
- If \(X\subseteq \mathbb R^3\), how many vectors does \(X\) have at most?

**Proposition 6.** *Let \(V\) be an euclidean space. Let \(X = \{v_1, v_2,
\ldots, v_k\}\) be an orthogonal set such that \(v_j\neq 0\) for all
\(j\in[1,\ldots,k]\). Then \(X\) is linearly independent.*

*Proof.*
\[\langle\alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_k v_k, v_j\rangle
= \alpha_j^2 \lVert v_j \rVert^2 = 0 \implies \alpha_j = 0\]

**Corollary 2.** *Let \(V\) be an euclidean space of dimension \(n\), and
let \(X = \{v_1, v_2, \ldots, v_k\}\) be an orthogonal set such that
\(v_j \neq 0\) for all \(j\in[1, k]\). Then \(k \leq n\).*

**Corollary 3.** *Let \(V\) be an euclidean space of dimension \(n\), and
let \(X = \{v_1, v_2, \ldots, v_n\}\) be an orthogonal set such that
\(v_j \neq 0\) for all \(j\in[1, n]\). Then \(X\) is a basis of \(V\).*

### Orthogonal complements of the subspaces of a real matrix⌗

**Proposition 7.** *Let \(A\) be a \(n \times k\) matrix with real elements.
Then, considering in \(\mathbb R^n\) and \(\mathbb R^k\) the usual inner
products we have:*^{2}

- \(L(A)^\perp = N(A)\)
- \(N(A)^\perp = L(A)\)
- \(C(A)^\perp = N(A^T)\)
- \(N( A^T )^\perp = C(A)\)

## Orthogonal Projections⌗

### Orthogonal bases and orthonormal bases⌗

A basis \(\mathcal{B}\) of an euclidean space \(V\) is said to be:

- An
**orthogonal basis**if it is an orthogonal set; - An
**orthonormal basis**if it is an orthogonal set, and all it’s elements have unitary norm.

Let \(x\in V\) some vector, and let \[(x)_\mathcal{B} = (\alpha_1, \alpha_2, \ldots, \alpha_n)\] be the coordinate vector of \(x\) in the basis \(\mathcal B\).

### Coordinate vector in an orthogonal basis \(\mathcal B\)⌗

\[\alpha_j = \frac{\langle x, b_j\rangle}{\lVert b_j\rVert^2}\]

### Coordinate vector in an orthonormal basis \(\mathcal B\)⌗

\[ \alpha_j = \langle x, b_j\rangle\]

**Question.** Will there always be an orthogonal and/or an orthonormal basis?

**Answer.** Yes -> Orthogonalization through Gram-Schmidt method.

### Orthogonal projections⌗

We define the **orthogonal projection of \(x\) over \(b_j\)** as the vector
\[\begin{aligned}\text{proj}_{b_j} x &= \frac{\langle x, b_j \rangle}{\lVert
b_j \rVert^2}b_j \\ &= \alpha_j b_j \end{aligned}\]

In a more general sense, given two vectors \(u\) and \(v\) from an euclidean
space \(V\), with \(v\neq 0\) the **orthogonal projection of \(u\) over
\(v\)** is the vector
\[\text{proj}_v u = \frac{\langle u, v \rangle}{\lVert v \rVert}^2 v\]

**Example.** Considering that \(\mathbb R^2\) is equipped with the canonical
basis \(\mathcal{E}_2 = (e_1, e_2)\), any vector \(u\in\mathbb R^2\) can
be expressed as a sum
\[\begin{aligned}u &= \text{proj}_{ e_1} u + \text{proj}_{ e_2} u \\ &=
u_W + u_{W^\perp}\end{aligned}\]
Where \(W\) is the \(x\) axis.

**Theorem 5.** *Let \(W\) be a linear subspace of some euclidean space
\(V\). All vectors \(u\) of \(V\) can be decomposed uniquely as*
\[u = u_W + u_{W^\perp}\]

*where \(u\in W\) and \(u_{W^\perp}\in E^\perp\).*

In these conditions, we say that \(V\) is the **direct sum** of \(W\) with
\(W^\perp\) and write
\[V = W \oplus W^\perp\]
Which, by definition, is to say:

- \(V = W + W^\perp\)
- \(W \cap W^\perp = \{0\}\)

We define the **orthogonal projection of \(u\) over \(W\)** as being the
vector \(u_W\).

If we consider that \(W\) is equipped with the ordered **orthogonal** basis
\(\mathcal{B} = (b_1, b_2, \ldots, b_k)\), we have
\[\text{proj}_W u = \text{proj}_{b_1} u + \text{proj}_{b_2} u + \cdots +
\text{proj}_{b_k} u\]

**Question.** How can we compute the vector \(u_{W^\perp}\) or, in other
words, \(\text{proj}_{W^\perp} u\)?

**Answer.**
\[\text{proj}_{W^\perp} u = u - u_W\]
or, if we consider that \(W^\perp\) is equipped with the ordered orthogonal
basis \(\mathcal{B}' = (b_1', b_2', \ldots, b_l')\), we have
\[\text{proj}_{W^\perp} u = \text{proj}_{b'_1} u + \text{proj}_{b'_2} u +
\cdots + \text{proj}_{b'_l} u\]

**Question.** What is the number \(l\) of vectors in the basis of \(\mathcal{B}'\)?

**Answer.** Assuming that \(V\) has dimension \(n\), we have \(l = n - k\)
since

- \(\mathcal{B} \cup \mathcal{B}'\) is linearly independent.
^{3} - Theorem 5 guarantees that \(\mathcal{B}\cup\mathcal{B}'\) generates \(V\).

Therefore \(\mathcal{B} \cup \mathcal{B}'\) is a basis of \(V\) and the solution becomes trivial.

## Distance from a point to a subspace & \(k\)-plane cartesian equations⌗

### Optimal approximation⌗

Given \(u\in V\) and some subspace \(W\) of \(V\) we hope to answer the following question:

Which element \(x\) of \(W\) is closest to \(u\)?

\[\begin{aligned}d(u, x)^2 = \lVert u - x\rVert^2 &= \lVert(u - \text{proj}_W u) + (\text{proj}_W u - x)\rVert^2 \\ &= \lVert u - \text{proj}_W u\rVert^2 + \lVert \text{proj}_W u - x\rVert^2 \qquad \text{(Pythagoras)}\\ &= \lVert \text{proj}_{W^\perp} u\rVert^2 + \lVert\text{proj}_W u - x\rVert^2\end{aligned}\]

Whereby we conclude that

The optimal approximation coincides with \(\text{proj}_W u\)

^{4}

With that, we define the **distance from \(u\) to a subspace \(W\)** as

\[d(u, W) = \lVert proj_{W^\perp} u \rVert\]

### \(k\)-plane cartesian equations⌗

A **\(k\)-plane** of \(\mathbb R^n\) is any subset \(S\) of \(\mathbb
R^n\) which can be expressed as

\[S = W + p\]

Where \(W\) is a subspace of \(\mathbb R^n\) with dimension \(k\) and \(p\) is an element of \(\mathbb R^n\). Depending on the dimension of \(W\), we have the following nomenclature:

- If \(k = 0\), \(S\) is said to be a
**point**. - If \(k = 1\), \(S\) is said to be a
**line**. - If \(k = 2\), \(S\) is said to be a
**plane**. - If \(k = n - 1\), \(S\) is said to be a
**hyperplane**.^{5}

Let \(x = (x_1, x_2, \ldots, x_n)\) be an elements of \(S\), there exists \(y\) in \(W\) such that

\[x = y + p\]

Or equivalently

\[y = x - p\]

The last equation show that, using vector, cartesian, or parametric equations of \(W\) we can easily obtain (substituting \(y\) for \(x-p\)) vector, cartesian, or parametric equations of \(S\), respectively.

Analogously, using the subspace \(W^\perp\) we can also obtain equations of \(S\). If \(B_{W^\perp} = (v_1, v_2, \ldots, v_{n-k})\) is a basis for the orthogonal complement of \(W\), with \(\text{dim} W = k\), we have \(x - p \in W\) or, equivalently

\[\underbrace{\begin{bmatrix} v^T_1 \\ v^T_2 \\ \vdots \\ v^T_{n-k} \end{bmatrix}}_{(n-k)\times n} \underbrace{\begin{bmatrix} x_1 - p_1 \\ x_2 - p_2 \\ \vdots \\ x_n - p_n \end{bmatrix}}_{n\times 1} = \underbrace{\begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}}_{(n-k)\times 1}\]

Defining the matrix \(A\) as

\[A = \begin{bmatrix}v^T_1 \\ v^T_2 \\ \vdots \\ v^T_{n-k} \end{bmatrix}\]

We obtain the homogeneous linear equation system \(A(x -p) = 0\). Consequently, from a vector equation of \(N(A)\), or cartesian equations of \(N(A)\), or parametric equations of \(N(A)\), we can obtain the corresponding equations of \(S\).

**Exercise.** Determine a vector equation, the cartesian equations, and the
parametric equations of the plane passing the point \(p = (1, 2, 0)\) which is
perpendicular to the line passing this same point with direction \(n=(5, 1, -2)\)

### Distance from a point to a \(k\)-plane⌗

Let \(S=W+p\) and consider a point \(q\in\mathbb R^n\). Given \(x\) in \(S\),

\[\begin{aligned}d(q, x) &= \lVert q - x \rVert \\ &= \lVert (q - p) + (p - x) \\ &= \lVert (q - p) - y \rVert \\ &= d(q-p, y) \\ \end{aligned}\]

The minimal value for this distance can be obtained for \(y =
\text{proj}_{W}(q - p)\), as previously described. We then define the
**distance from point \(q\) to the plane \(S\)** as

\[\begin{aligned}d(q, S) &= d(q - p, W) \\ &= \lVert \text{proj}_{W^\perp}(q - p )\rVert\end{aligned}\]

**Exercise.** Compute the distance from \((3, 2, -1)\) to the plane \(S\)
from the previous exercise.

Note that \(tr(B^T A) = tr(A^T B)\), which allows us to define \[\langle A, B\rangle = tr(A^T B) \] ↩︎

I don’t know whether these function names are right in English. IIRC, from my Portuguese notes, L(A) is the space of the lines of a matrix, C(A) is the space of the columns, and N(A) is the kernel. ↩︎

Because it is orthogonal. ↩︎

The closest point to \(u\) in \(W\) is \(\text{proj}_W u\). ↩︎

If \(k = n\), \(S = \mathbb R^n\). ↩︎