## 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 $$\alpha \in \mathbb R$$,

1. $$\langle x,y \rangle = \langle y, x \rangle$$
2. $$\langle \alpha x, y\rangle = \alpha\langle x, y\rangle$$
3. $$\langle x+y, z\rangle = \langle x, z\rangle + \langle y, z\rangle$$
4. $$\langle x, x\rangle \geq 0 \wedge (\langle x, x\rangle = 0 \implies x = 0)$$

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

### Examples⌗

1. 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}\
2. 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

1. $$\lvert x\rVert\geq 0\qquad \text{and}\qquad \lVert x\rVert \iff x = 0$$
2. $$\lVert \alpha x\rVert = \lvert\alpha\rvert\lVert x\rVert$$
3. $$\lVert x + y\rVert\leq\lVert x\rVert + \lVert y\rVert \qquad\qquad \text{(triangle inequality)}$$

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.

1. The expression $\langle x, y\rangle = y^T Ax$ defines an inner product in $$\mathbb R^n$$
2. $$A$$ 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 $$\alpha \in \mathbb C$$

1. $$\langle x, y\rangle = \overline{\langle y, x\rangle}$$
2. $$\langle \alpha x, y\rangle = \alpha\langle x,y\rangle$$
3. $$\langle x + y, z\rangle = \langle x, z\rangle + \langle y, z\rangle$$
4. $$\langle x, x\rangle \geq 0 \wedge (\langle x, x\rangle = 0 \implies x = 0)$$

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.

1. The expression $\langle x, y\rangle = \overline{y}^T Ax$ defines an inner product in $$\mathbb C^n$$
2. 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 $$V$$.

1. $$W\cap W^\perp = 0$$
2. $$W^{\perp\perp} = W$$

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

1. $$L(A)^\perp = N(A)$$
2. $$N(A)^\perp = L(A)$$
3. $$C(A)^\perp = N(A^T)$$
4. $$N( A^T )^\perp = C(A)$$

## 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

1. $$\mathcal{B} \cup \mathcal{B}’$$ is linearly independent.3
2. 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.

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

2. 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. ↩︎

3. Because it is orthogonal. ↩︎

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

5. If $$k = n$$, $$S = \mathbb R^n$$. ↩︎