OpenLLM-France/wikimedia huggingface dataset:
# Permutations of Boolean functions

Analogous to hard and soft properties of Boolean functions, there are also hard and soft permutations.
A permutation is hard, when the domain is the infinite set of all Boolean functions. An example is the map from a BF to its complement.
A permutation is soft, when the domain is the finite set of BF with a given arity. An example is the map from a BF to its Zhegalkin twin.
Interesting permutations of BF are often Walsh permutations, which correspond to an invertible binary matrix of size $2^{arity}$.
c:Category:8-ary Walsh functions in octeract matrix

## Zhegalkin permutation

This might be the most important permutation of Boolean functions. It is soft. A Boolean function can be represented by its truth table or its Zhegalkin index. As a consequence, every other permutation of Boolean functions can be represented by four different permutation of integers: Between truth tables, between Zhegalkin indices, and from one to the other. Mathematically the one between Zhegalkin indices is probably most important. In the hard case, it is an infinite permutation.

## Mentors of Boolean functions

The mentor permutation might be called semi-hard, because it is an infinite permutation between pairs of complements.

## gradual sharpness

T to T: (47803, 47800, 47549, 47537, 35996, 36012, 32960, 32768, 1, 769, 13617, 14641, 36253, 48541, 7517, 56669) T to Z: (65293, 43608, 52331, 34863, 61492, 41060, 49216, 32768, 65535, 43775, 52399, 34991, 61643, 41163, 49291, 32907) Z to T: (65535, 21845, 52428, 56797, 61680, 27756, 49344, 23901, 65280, 23280, 52224, 58476, 61440, 24768, 49152, 56669) Z to Z: (1, 3, 4, 11, 16, 36, 64, 139, 256, 528, 1024, 2084, 4096, 8256, 16384, 32907) A127804
# Sum norm/Norm/Exercise

Show that the sum norm on ${\mathbb {K} }^{n}$ is a norm.
# Vector/R^3/Norming/Several norms/Exercise

Determine, for the vector
    ${\begin{pmatrix}-6\\10\\7\end{pmatrix}}\in \mathbb {R} ^{3}\,,$
the corresponding normed vector with respect to the Euclidean norm, the maximum norm, and the sum norm.
# C Sharp

C Sharp (C#) is a multi-paradigm programming language encompassing strong typing, imperative, declarative, functional, procedural, generic, object-oriented (class-based), and component-oriented programming disciplines. It was developed by Microsoft within its .NET initiative and later approved as a standard by Ecma (ECMA-334) and ISO (ISO/IEC 23270:2006). C# is one of the programming languages designed for the Common Language Infrastructure.

## Prerequisites

* Introduction to Programming
* Introduction to Object Oriented Programming
* Microsoft Visual Studio

## See Also

* Computer Programming
* csharp.net: Tutorials
* C# Tutorial Online
# Segment display/Seven-segment display

A nine-segment display (SSD), or nine-segment indicator, is a form of electronic display device for displaying decimal numerals that is an alternative to the more complex dot matrix displays. Nine-segment displays are widely used in digital clocks, electronic meters, basic calculators, and other electronic devices that display numerical information.

## Examples

Media player, 4-DIGIT decoder

### Digits

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
W

### alphabet uppercase

| A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z | Error | Ƃ | Ƌ 4 |

### alphabet lowercase

| a | b | c | d | e | f | g | h | i | j | k | l | m | n | o | p | q | r | s | t | u | v | w | x | y | z |

### Greek alphabet

| Α | Β | β | Γ | γ | Δ | δ | Ε | Ζ | ζ | Θ | Η | η | Ι | ι | Κ | κ | Λ | λ | Μ | μ | Ν | ν | Ξ | ξ | Ο | ο | Π | π | Ρ | Σ | σ | Τ | τ | Υ | υ | Φ | Χ | χ | Ψ | Ω | ω | Ϝ | Ϲ | Ͻ | 1 |

### Cyrillic alphabet

| А | а | Ж | ж | В | в | Д | д | Е | е | Н | н | Г | г | Ь | ь | Ц | ц | Б | б | Ч | ч | М | м | Л | л | И | и | Й | й | Ъ | ъ | Ю | ю | О | о | Р | р | С | с | Ѕ | ѕ | Т | т | У | у | Ш | ш | Щ | щ | К | к | З | з | Э | э | Х | х | Ф | ф | П | п | Я | я |

### Punctuation

| + | - | × | ÷ | = | % | _ | ! | @ | # | $ | / | ^ | & | * | ( | ) | £ | € | ' | " | : | ; | , | ? | ` | ~ | \ | | | < | > | [ | ] |
* ↑ Wikipedia: nine-segment display
# Maximum norm/No inner product/Exercise

Let $n\geq 2$. Show that the norm $\Vert {x}\Vert :=\operatorname {max} \{\vert {x_{i}}\vert :1\leq i\leq n\}$ on $\mathbb {R} ^{n}$ does not come from an inner product $\left\langle -,-\right\rangle$.
# Normed vector space/C/Real/Exercise

Let $V$ be a normed vector space over $\mathbb {C}$. Show that $V$, considered as a real vector space, is, with this norm, also a normed vector space.
# Several distances/(3 by 4,-1) and (2,1 by 5)/Exercise

We consider the two points $P=\left({\frac {3}{4}},\,-1\right)$ and $Q=\left(2,\,{\frac {1}{5}}\right)$ in $\mathbb {R} ^{2}$. Determine the distance between these points with respect to the
a) the Euclidean metric
b) the sum metric,
c) the maximum metric.
d) Compare these different distances according to their size.
# Hamming-distance/Product set/Metric/Examples/Exercise

Let $A$ denote a set, $n\in \mathbb {N} _{+}$, and $M=A^{n}$ the $n$-fold product set of $A$ with itself.
a) Show that
    $d(x,y)=d((x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n})):={\#\left({\left\{i\mid x_{i}\neq y_{i}\right\}}\right)}\,$
defines a metric on $M$.
b) For $A=\{a,b,c\}$, and $n=4$, determine the distance $d((a,a,b,c),(c,a,b,a))$.
c) For $A=\{a,b,c\}$, and $n=3$, list all elements in the open ball $[[:Template:Op:Openball]]$.
# JavaScript/Smooth counting

Smooth counting means that a number will increase steeply at the beginning and then will level off. This can be accomplished with the help of mathematical functions.
# Train stations/Shortest connection/Exercise

Let $M$ be the set of all train stations in Germany. For $a,b\in M$, define
    $d(a,b)$
to be the (in terms of scheduled time) shortest connection from $a$ to $b$. Is this a metric?
# Coordinate axes/Metric space/Exercise

Let $A\subseteq \mathbb {R} ^{2}$ be the union of the $x$-axis and the $y$-axis.
a) Define the distance on $A$ given by shortest connection between the two points $P,Q\in A$ by a path on $A$.
b) Show that this is a metric.
c) Does there exist a norm on $\mathbb {R} ^{2}$ such that the restriction of the corresponding metric coincides with the given connecting metric?
# Affine space/Normed vector space/Metric space/Exercise

Let $V$ be a normed ${\mathbb {K} }$-vector space, and let $E$ denote an affine space over $V$. Show that $E$ becomes a metric space by setting
    $d(P,Q):=\Vert {\overrightarrow {PQ}}\Vert \,.$
# Bounded functions/Supremum norm/Vector space/Exercise

Let $T$ be a set, and let
    $M={\left\{f:T\rightarrow \mathbb {C} \mid \Vert {f}\Vert _{T}<\infty \right\}}\,$
denote the set of bounded complex-valued functions on $T$. Show that $M$ is a complex vector space.
# Bounded functions/Supremum norm/Norm properties/Exercise

Let $T$ be a set, and let
    $M={\left\{f:T\rightarrow \mathbb {C} \mid \Vert {f}\Vert _{T}<\infty \right\}}\,$
be the set of all bounded complex-valued functions on $T$. Show that the supremum norm on $M$ fulfills the following properties.
* $\Vert {f}\Vert \geq 0$ for all $f\in M$.
* $\Vert {f}\Vert =0$ if and only if $f=0$.
* For $\lambda \in \mathbb {C}$, and $f\in M$, we have
    $\Vert {\lambda f}\Vert =\vert {\lambda }\vert \cdot \Vert {f}\Vert \,.$
* For $g,f\in M$, we have
    $\Vert {g+f}\Vert \leq \Vert {g}\Vert +\Vert {f}\Vert \,.$
# Bounded mappings/Euclidean space/Supremum norm/Norm/Exercise

Let $T$ be a set, and let $E$ be a Euclidean vector space. Let
    $M={\left\{f:T\rightarrow E\mid \Vert {f}\Vert _{T}<\infty \right\}}\,$
be the set of all bounded mappings from $T$ to $E$. Show that the supremum norm on $M$ is a norm.
# Bounded mappings/Euclidean space/Uniform convergence and supremum convergence/Exercise

Let $T$ be a set, and let $E$ be a Euclidean vector space. Let
    $M={\left\{f:T\rightarrow E\mid \Vert {f}\Vert _{T}<\infty \right\}}\,$
be the set of all bounded mappings from $T$ to $E$. Show that a sequence ${\left(f_{n}\right)}_{n\in \mathbb {N} }$ in $M$ converges uniformally to $f\in M$ if and only if this sequence converges to $f$ in $M$ with respect to the supremum norm.
# MDLD/inner product (real)

Start=inner product (real) Target=Vector space/K/Inner product/Definition
# MDLD/inner product (complex)

Start=inner product (complex) Target=Vector space/K/Inner product/Definition
# MDLD/complex vector space

Start=complex vector space Target=Vector space/Introduction/Section
# MDLD/maximum norm

Start=maximum norm Target=K^n/Maximum norm/Example
# MDLD/sum norm

Start=sum norm Target=K^n/Sum norm/Example
# Normed vector space/K/Introduction/Section

Due to fact, the norm associated to an inner product is a norm in the sense of the following definition. In particular, a vector space with an inner product is a normed vector space.

## Definition

Let $V$ be a ${\mathbb {K} }$-vector space. A mapping
    $\Vert {-}\Vert \colon V\longrightarrow \mathbb {R} ,v\longmapsto \Vert {v}\Vert ,$
is called norm, if the following properties hold.
* We have $\Vert {v}\Vert \geq 0$ for all $v\in V$.
* We have $\Vert {v}\Vert =0$ if and only if $v=0$.
* For $\lambda \in {\mathbb {K} }$ and $v\in V$, we have
    $\Vert {\lambda v}\Vert =\vert {\lambda }\vert \cdot \Vert {v}\Vert \,.$
* For $v,w\in V$, we have
    $\Vert {v+w}\Vert \leq \Vert {v}\Vert +\Vert {w}\Vert \,.$

## Definition

A ${\mathbb {K} }$-vector space is called a normed vector space if a norm
$\Vert {-}\Vert$
On a Euclidean vector space, the norm given via the the inner product is also called the Euclidean norm. For $V=\mathbb {R} ^{n}$, endowed with the standard inner product, we have
    $\Vert {v}\Vert ={\sqrt {\sum _{i=1}^{n}v_{i}^{2}}}\,.$

## Example

In ${\mathbb {K} }^{n}$, taking
    $\Vert {x}\Vert _{\rm {max}}:={\max {\left(\vert {x_{i}}\vert ,i=1,\ldots ,n\right)}}\,,$
a norm is defined, which is called the maximum norm.

## Example

In ${\mathbb {K} }^{n}$, taking
    $\mid \mid \!x\!\mid \mid _{\operatorname {sum} }:=\sum _{i=1}^{n}\vert {x_{i}}\vert \,,$
defines a norm, which is called the sum norm.
For a vector $v\in V$, $v\neq 0$, in a normed vector space $V$, the vector ${\frac {v}{\Vert {v}\Vert }}$ is called the corresponding normalized. Such a normalized vector has norm $1$. Passing to the normalized vector is also called normalization.
# MDLD/Euclidean norm

Start=Euclidean norm Target=Normed vector space/K/Introduction/Section
# MDLD/normed (vs)

Start=normed (vs) Target=Vector space/K/Norm/Definition
# MDLD/metric

Start=metric Target=Metric/Metric space/Definition
# MDLD/complex-valued

Start=complex-valued Target=Complex numbers/Pairs/Definition
# Vector space/K/Inner product/Finite-dimensional/Orthogonal complement/Structural properties/Fact/Proof/Exercise

Let $V$ be a ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Show the following statements.
* For a linear subspace $U\subseteq U'\subseteq V$, we have
    $U^{\perp }\supseteq U'^{\perp }\,.$
* We have $0^{\perp }=V$ and $V^{\perp }=0$.
* Let $V$ be finite-dimensional. Then we have
    ${\left(U^{\perp }\right)}^{\perp }=U\,.$
* Let $V$ be finite-dimensional. Then we have
    $\dim _{K}{\left(U^{\perp }\right)}=\dim _{K}{\left(V\right)}-\dim _{K}{\left(U\right)}\,.$
# Metric space/Open ball/Definition

Let $(M,d)$ be a metric space, $x\in M$, and $\epsilon >0$ a positive real number. Then
    $U{\left(x,\epsilon \right)}={\left\{y\in M\mid d(x,y)<\epsilon \right\}}\,$
is called the open ball about $x$ with radius $\epsilon$.
# Vector space/K/Inner product/Norm equality/Exercise

Let $V$ be a ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Show that, for $u,v\in V$, the equality $\Vert {v+u}\Vert =\Vert {v-u}\Vert$ holds if and only if $\left\langle v,u\right\rangle =0$ holds.
# Linear algebra (Osnabrück 2024-2025)/Vector space/K/Inner product/Norm equality/Exercise/Exercisereferencenumber

Exercise 32.1
# R^3/Example vectors/Orthogonal/1/Exercise

Determine which of the following vectors in $\mathbb {R} ^{3}$ are orthogonal to each other with respect to the standard inner product.
    ${\begin{pmatrix}6\\1\\5\end{pmatrix}},\,{\begin{pmatrix}3\\-8\\-2\end{pmatrix}},\,{\begin{pmatrix}0\\-1\\4\end{pmatrix}},\,{\begin{pmatrix}-5\\4\\-1\end{pmatrix}}.$
# C^2/Example vectors/Orthogonal/1/Exercise

Determine which of the following vectors in $\mathbb {C} ^{2}$ are orthogonal to each other with respect to the standard inner product.
    ${\begin{pmatrix}6-3{\mathrm {i} }\\-{\mathrm {i} }\end{pmatrix}},\,{\begin{pmatrix}4-7{\mathrm {i} }\\-9-5{\mathrm {i} }\end{pmatrix}},\,{\begin{pmatrix}{\frac {1}{3}}{\mathrm {i} }\\2+{\mathrm {i} }\end{pmatrix}},\,{\begin{pmatrix}{\frac {1}{3}}\\-1-2{\mathrm {i} }\end{pmatrix}},\,{\begin{pmatrix}{\frac {1}{3}}\\-1+2{\mathrm {i} }\end{pmatrix}}.$
# Inner product/Component and values fixed/Affine subspace/Exercise

Let $V$ be a ${\mathbb {K} }$-vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let $v\in V$ be a fixed vector, and $a\in {\mathbb {K} }$. Show that
    ${\left\{u\in V\mid \left\langle u,v\right\rangle =a\right\}}$
is an affine subspace of $V$.
# Pythagorean Theorem/Status/Exercise

Discuss the relationship between the Pythagorean theorem in the sense of fact and its elementary-geometric version.
# Linear algebra (Osnabrück 2024-2025)/Vector space with inner product/Theorem of Pythagoras/Fact/Factreferencenumber

Theorem 32.3
# Inner product/K/Orthogonal complement/Linear subspace/Exercise

Let $V$ be a vector space over ${\mathbb {K} }$, endowed with an inner product $\left\langle -,-\right\rangle$, and let $U\subseteq V$ be a linear subspace. Show that the orthogonal complement is also a linear subspace of $V$.
# Linear algebra (Osnabrück 2024-2025)/Inner product/K/Orthogonal complement/Linear subspace/Exercise/Exercisereferencenumber

Exercise 32.6
# Orthogonal complement/K/Test on generating system/Exercise

Let $V$ be a vector space over ${\mathbb {K} }$, endowed with an inner product $\left\langle -,-\right\rangle$, and let $U\subseteq V$ be a linear subspace. Let $u_{i}$, $i\in I$, be a generating system of $U$. Show that a vector $v\in V$ belongs to the orthogonal complement $U^{\perp }$ if and only if
    $\left\langle v,u_{i}\right\rangle =0\,$
holds for all $i\in I$.
# Linear algebra (Osnabrück 2024-2025)/Orthogonal complement/K/Test on generating system/Exercise/Exercisereferencenumber

Exercise 32.7
# MDLD/orthogonal (R)

Start=orthogonal (R) Target=Vector space/K/Inner product/Orthogonal/Definition
# MDLD/standard inner product (C)

Start=standard inner product (C) Target=C^n/Standard inner product/Definition
# Sequence Balm

families :
| introverted: Mint | blunt: Balm |
| extroverted: DubSage | sharp: Sage |
| total: Thyme ||
${\text{Balm}}(n)={\frac {2^{2^{n}}+(2^{2^{n-1}{\color {Red}+1}})(2^{n}-1)}{2^{n{\color {Red}+1}}}}$

## Sequences Balm$_{(Drop)}$

| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
| 🌊 Balm (A000133) | 1 | 2 | 5 | 30 | 2288 | 67172352 | 144115192303714304 | 1329227995784915891206435945914040320 |
| 💧 BalmDrop | 1 | 1 | 3 | 25 | 2258 | 67170064 | 144115192236541952 | 1329227995784915891062320753610326016 |
# Properties of Boolean functions/soft

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Soft properties depend on the arity of the BF.

## twin

The Zhegalkin twin of a BF is its Zhegalkin index interpreted as a truth table of length 2$^{arity}$.

## half-complement

The half-complement of a BF is its XOR with x$_{arity−1}$. See e.g. the half-complements of $A\land B$:
* arity 3: $(A\land B)\oplus C$ 0001 0001 XOR 0000 1111 = 0001 1110
* arity 4: $(A\land B)\oplus D$ 0001 0001 0001 0001 XOR 0000 0000 1111 1111 = 0001 0001 1110 1110

## ultra equivalence classes

Families and clans can be merged with their complements, which creates super-families and super-clans. Further merging them with their half-complements creates ultra-families and ultra-clans. This family is a complete ultra-family: 1100 1010 (So its matrix of truth tables is mirror symmetric.) (Factions do not have a unique half-complement, so there are no ultra-factions.)

## tribe

For a given arity the tribe is the next biggest equivalence class after the ultra-clan. Blunt BF belong to tribes denoted by their consul weight. All sharp BF belong to a tribe on their own.
tribes for arities 1, 2, 3 :

## principalities and dominions

A principality is a set of n-ary BF whose (n+1)-ary noble equivalents form a faction. Dominions are closely related.
# Corner in room/Plane diagonals/Exercise

We consider a corner in a (rectangular) room. Do the three diagonal vectors in the two adjacent walls and on the ground of length $1$ form an orthonormal basis?
# Properties of Boolean functions/soft/binary

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Soft properties depend on the arity of the BF. Binary properties have the values true or false, so they partition all BF into a subset and the rest.
number of true places in the truth table of length 2$^{arity}$ (hard weight is a fraction)

## related to parity

* BF is sharp/blunt iff weight of the truth table is odd/even. This is an important property of families.
* rude = odd XOR sharp
* rough = odious XOR sharp
* acute = ugly XOR sharp The opposite is obtuse.
images for arity 3 :
| | sharp | rude | rough | acute | |
| truth tables | 255 1111 1111 | 254 0111 1111 | 127 1111 1110 | 126 0111 1110 | Sharp, acute, rough and rude are the four soft foibles. |
| Zhegalkin indices | 128 0000 0001 | 129 1000 0001 | 127 1111 1110 | 126 0111 1110 | Sharp, acute, rough and rude are the four soft foibles. |

## symmetric

Symmetric BF do not depend on the order of their inputs. (They are the unique members of their factions.)

## noble

Noble BF are their own Zhegalkin twins.
# Properties of Boolean functions/hard/integer

Properties of Boolean functions :
| hard | soft |
| binary | binary |
| integer | integer |
Hard properties can be assigned to a BF, without referencing its arity. Integer properties have all non-negative integers as values.

## valency ≤ adicity ≤ arity

Valency is the number of atoms, i.e. the number of arguments acutally used. It is the number of circles in the Euler diagram.
Adicity follows from the biggest atom. 2$^{adicity}$ is the required length of the truth table, or the period length of the infinite truth table.
In this project arity is not a property of a BF. The term is used in two slightly different ways:
* arity n as a shorthand for adicity ≤ n (as in: the 16 2-ary Boolean functions)
* arity n as an argument to get a finite truth table of length $2^{n}$ E.g. $B\land C$ can be shown as 3-ary 0000 0011 or as 4-ary 0000 0011 0000 0011.
This can be explained in terms of object-oriented programming: Valency and adicity are attributes of a BF class. Arity is an argument to some of its methods.

## strength

2$^{strength}$ = family size

## gravity and depth

2$^{gravity}$ and 2$^{depth}$ are the cardinalities of legion and cohort. The result for the contradiction is infinity. Soft equivalents have been defined to avoid this problem. Soft gravity is twin valency. Soft depth is related to strength.
# Orthonormal basis/Complex/Real/Exercise

Let $V$ be a finite-dimensional complex vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let $u_{1},\ldots ,u_{n}$ be an orthonormal basis of $V$. Show that
    $u_{1},{\mathrm {i} }u_{1},u_{2},{\mathrm {i} }u_{2},\ldots ,u_{n},{\mathrm {i} }u_{n}$
is an orthonormal basis of the real vector space $V$ with respect to the corresponding real inner product.
# Inner product/Orthonormal basis/Standard evaluation/Exercise

Let $V$ be a real vector space, endowed with an inner product $\left\langle -,-\right\rangle$. Let $u_{i}$, $i\in I$, denote an orthonormal basis of $V$. Show that for vektors $v=\sum _{i\in I}a_{i}u_{i}$ and $w=\sum _{i\in I}b_{i}u_{i}$, the equality
    $\left\langle v,w\right\rangle =\sum _{i\in I}a_{i}b_{i}\,$
holds.
# Linear algebra (Osnabrück 2024-2025)/Inner product/Orthonormal basis/Standard evaluation/Exercise/Exercisereferencenumber

Exercise 32.13
# Exponential functions/Frequencies/01/Orthonormal system/Fact/Proof/Exercise

Show that the functions
    $f_{m}\colon [0,1]\longrightarrow \mathbb {C} ,$
defined by
    $f_{m}(x):=e^{2\pi {\mathrm {i} }mx}\,$
for $m\in \mathbb {Z}$, form, in the space of continuous functions from $[0,1]$ to $\mathbb {C}$, an orthonormal system with respect to the inner product defined by
    $\left\langle f,g\right\rangle :=\int _{0}^{1}f{\overline {g}}dx\,.$
Use basic facts about the complex exponential function.
# Orthonormalization/(0,1,1),(1,0,1),(1,1,0)/Exercise

Apply Gram-Schmidt orthonormalization to the basis
    ${\begin{pmatrix}0\\1\\1\end{pmatrix}},\,{\begin{pmatrix}1\\0\\1\end{pmatrix}},\,{\begin{pmatrix}1\\1\\0\end{pmatrix}}$
of $\mathbb {R} ^{3}$.
# Orthonormalization/(2,1,0),(1,0,-1),(0,0,3)/Exercise

Apply Gram-Schmidt orthonormalization to the basis
    ${\begin{pmatrix}2\\1\\0\end{pmatrix}},\,{\begin{pmatrix}1\\0\\-1\end{pmatrix}},\,{\begin{pmatrix}0\\0\\3\end{pmatrix}}$
of $\mathbb {R} ^{3}$.
# Inner product/(-1 2 3), (2 -4 5), (7 3 1)/Orthonormal basis/Exercise

Apply Gram-Schmidt orthonormalization to the basis
    ${\begin{pmatrix}-1\\2\\3\end{pmatrix}},\,{\begin{pmatrix}2\\-4\\5\end{pmatrix}},\,{\begin{pmatrix}7\\3\\1\end{pmatrix}}$
of $\mathbb {R} ^{3}$, endowed with the standard inner product.
# Orthonormal basis/R^3/(1,1,1)/Exercise

Establish an orthonormal basis of $\mathbb {R} ^{3}$ that contains a multiple of ${\begin{pmatrix}1\\1\\1\end{pmatrix}}$.
# Inner product/Kernel of 317/Orthonormal basis/Exercise

We endow $\mathbb {R} ^{3}$ with the standard inner product. Let $U\subseteq \mathbb {R} ^{3}$ be the kernel of the linear mapping
    $\mathbb {R} ^{3}\longrightarrow \mathbb {R} ,(x,y,z)\longmapsto 3x+y+7z,$
endowed with the restricted inner product. Determine an orthonormal basis for $U$.
# Vector space/R/Finite-dimensional/Inner product/Isomorphy to dual space/Exercise

Let $V$ be a Euclidean vector space. Show that
    $V\longrightarrow {V}^{*},v\longmapsto {\left(w\mapsto \left\langle v,w\right\rangle \right)},$
gives an isomorphy between $V$ and its dual space ${V}^{*}$.
# Vector space/K/Inner product/Finite-dimensional/Orthogonal complement/Structural properties/Relation with dual space/Exercise

Prove fact, using exercise and fact.
# Linear algebra (Osnabrück 2024-2025)/Vector space/R/Finite-dimensional/Inner product/Isomorphy to dual space/Exercise/Exercisereferencenumber

Exercise 32.25
# Inner product/Kernel of 4 -3 2 -5/Orthonormal basis/Exercise

We endow $\mathbb {R} ^{4}$ with the standard inner product. Let $U\subseteq \mathbb {R} ^{4}$ be the kernel of the linear mapping
    $\mathbb {R} ^{4}\longrightarrow \mathbb {R} ,(x,y,z,w)\longmapsto 4x-3y+2z-5w.$
Determine an orthonormal basis for $U$.
# Orthonormalization/C^2/1/Exercise

Apply Gram-Schmidt orthonormalization to the basis
    ${\begin{pmatrix}4-{\mathrm {i} }\\3\end{pmatrix}},\,{\begin{pmatrix}{\mathrm {i} }\\2-{\mathrm {i} }\end{pmatrix}}$
of $\mathbb {C} ^{2}$.
# Orthonormal basis extending theorem/Formulate and proof/Exercise

Formulate and prove the "orthonormal basis completion theorem“.
# Vector space/Inner product/Orthogonal complement/Intersection and sum/Exercise

Let $V$ be a ${\mathbb {K} }$-vector space, endowed with an inner product, and let $U_{1},U_{2}\subseteq V$ denote linear subspaces. Show that for the orthogonal complements, the equality
    ${\left(U_{1}\cap U_{2}\right)}^{\perp }=U_{1}^{\perp }+U_{2}^{\perp }\,$
holds.
# Orthogonal projection/R^3/Line/1/Exercise

Determine the value of the vector ${\begin{pmatrix}3\\-5\\-3\end{pmatrix}}\in \mathbb {R} ^{3}$ under the orthogonal projection to the line gernerated by ${\begin{pmatrix}2\\4\\9\end{pmatrix}}$.
# Orthogonal projection/Transitivity/Exercise

Let $V$ be a finite-dimensional ${\mathbb {K} }$-vector space, endowed with an inner product. Let
    $U\subseteq W\subseteq V\,$
denote linear subspaces. Let $p_{U}^{V}$ denote the orthogonal projection from $V$ to $U$. Show
    $p_{U}^{V}=p_{U}^{W}\circ p_{W}^{V}\,.$