OpenLLM-France/wikimedia huggingface dataset:
# Affine mapping/Basis/Image points/Mapping properties/Exercise

Let $E$ and $F$ be affine spaces over the field $K$, let $P_{1},\ldots ,P_{n}\in E$ be an affine basis of $E$, and let $Q_{1},\ldots ,Q_{n}\in F$ denote points. Let
    $\psi \colon E\longrightarrow F$
be the corresponding affine-linear mapping with
    $\psi (P_{i})=Q_{i}\,.$
Show the following statements.
a) $\psi$ is bijective if and only if $Q_{1},\ldots ,Q_{n}$ is an affine basis of $F$.
b) $\psi$ is injective if and only if $Q_{1},\ldots ,Q_{n}$ are
affinely independent.
c) $\psi$ is surjective if and only if $Q_{1},\ldots ,Q_{n}$ is an affine generating system of $F$.
# Affine mapping/Linear part/Identity/Translation/Exercise

Let
    $\psi \colon E\longrightarrow E$
be an affine mapping on the affine space $E$. Show that the linear part $\psi _{0}$ is the identity if and only if $\psi$ is a translation.
# Affine mapping/Affine subspace/Image/Exercise

Let
    $\varphi \colon E\longrightarrow F$
be an affine-linear mapping between the affine spaces $E$ and $F$ over $K$. Show that, for every affine subspace $H\subseteq E$, the image $\varphi (H)$ is an affine subspace of $F$.
# Affine space/Finite point family/Mapping to affine hyperplane/Exercise

Let $E$ be an affine space, and let $P_{1},\ldots ,P_{n}$ be a finite family of points in $E$. Let
    $F={\left\{\left(a_{1},\,\ldots ,\,a_{n}\right)\in K^{n}\mid \sum _{i=1}^{n}a_{i}=1\right\}}\subset K^{n}\,.$
Show that the assignment
    $\left(a_{1},\,\ldots ,\,a_{n}\right)\longmapsto \sum _{i=1}^{n}a_{i}P_{i}$
defines a well-defined affine-linear mapping from $F$ to $E$.
# Affine spaces/Dimension and isomorphic/Exercise

Let $E$ and $F$ be affine spaces over the field $K$. Show that the spaces are isomorphic if and only if their dimensions coincide.
# Affine spaces/Product space/Projection/Exercise

Let $E$ and $F$ be affine spaces over the field $K$. Show that the
projections
    $E\times F\longrightarrow E$
and
    $E\times F\longrightarrow F$
are affine mappings.
# Affine space/Affine line/Description as preimage/1/Exercise

Describe the affine line
    $G={\left\{{\begin{pmatrix}6\\2\\3\end{pmatrix}}+s{\begin{pmatrix}-2\\5\\4\end{pmatrix}}\mid s\in \mathbb {R} \right\}}\,$
as the preimage over $(1,0)$ of an affine mapping $\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R} ^{2}$.
# Affine space/Affine plane/Description as preimage/1/Exercise

Describe the affine plane
    $E={\left\{{\begin{pmatrix}3\\1\\4\end{pmatrix}}+s{\begin{pmatrix}2\\7\\-6\end{pmatrix}}+t{\begin{pmatrix}-1\\5\\1\end{pmatrix}}\mid s,t\in \mathbb {R} \right\}}\,$
as the preimage over $1$ of an affine mapping $\psi \colon \mathbb {R} ^{3}\rightarrow \mathbb {R}$.
# Affine basis/Affinely independent generating system/Exercise

Let $E$ be an affine space over the $K$-vector space $V$, and let $P_{1},\ldots ,P_{n}$ be a finite family of points from $E$. Show that these points form an affine basis of $E$ if and only if they are affinely independent, and they are an affine generating system for $E$.
# R^3/Vector/Orthogonal space/1/Exercise

Determine a basis of the orthogonal space to $\mathbb {R} {\begin{pmatrix}5\\-4\\1\end{pmatrix}}\subseteq \mathbb {R} ^{3}$.
# Bidual/Finite-dimensional/Orthogonal space/Exercise

Let $V$ be a finite-dimensional $K$-vector space, with its dual space ${V}^{*}$ and the bidual ${{V}^{*}}^{*}$. Let $F\subseteq {V}^{*}$ be a linear subspace. Show that the two orthogonal spaces $F^{\perp }\subseteq V$ (in the sense of definition) and $F^{\perp }\subseteq {{V}^{*}}^{*}$ (in the sense of definition) are equal via the natural identification of the space and its bidual.
# Linear mapping/To finite-dimensional/Representation with linear forms/Fact

Let $K$ be a field and let $V$ and $W$ be vector spaces over $K$, where $W$ is finite-dimensional. Let
    $\varphi \colon V\longrightarrow W$
denote a linear mapping.
$w_{1},\ldots ,w_{n}\in W$ and linear forms $f_{1},\ldots ,f_{n}$ on $V$ such that
    $\varphi =f_{1}w_{1}+f_{2}w_{2}+\cdots +f_{n}w_{n}\,$
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/To finite-dimensional/Representation with linear forms/Fact/Factreferencenumber

Lemma 15.10
# Dual space/Linear forms/Linearly independent/Orthogonal space/Exercise

Let $V$ be a finite-dimensional $K$-vector space of dimension $n$, let $f_{1},\ldots ,f_{r}\in {V}^{*}$ be linear forms, and set
    $F=\langle f_{1},\ldots ,f_{r}\rangle \subseteq {V}^{*}\,.$
Show that these linear forms are linearly independent if and only if
    Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikiversity.org/v1/":): {\displaystyle {} \dim_{ K } { \left( F^{ \perp } \right) } = n-r \, }
holds.
# Vector spaces/Dual mapping/Total assignement/Linearity/Exercise

Let $K$ be a field, and let $V$ and $W$ denote $K$-vector spaces. Show that the mapping
    $\operatorname {Hom} _{K}{\left(V,W\right)}\longrightarrow \operatorname {Hom} _{K}{\left({W}^{*},{V}^{*}\right)},\varphi \longmapsto \varphi ^{*},$
which assigns to a linear mapping its dual mapping, is linear.
# Direct sum/N/Sum/No evaluation/Exercise

Let
    $V=\mathbb {R} ^{(\mathbb {N} )}\,$
be the countably direct sum of $\mathbb {R}$ with itself, with the basis $e_{n}$, $n\in \mathbb {N}$. Let $p_{k}$, $k\in \mathbb {N}$, be the projections
    $\mathbb {R} ^{(\mathbb {N} )}\longrightarrow \mathbb {R} ,\sum _{n\in \mathbb {N} }a_{n}e_{n}\longmapsto a_{k}.$
a) Show that
    $\varphi \colon V\longrightarrow \mathbb {R} ,\sum _{n\in \mathbb {N} }a_{n}e_{n}\longmapsto \sum _{n\in \mathbb {N} }a_{n},$
is a linear form on $V$, which is not a linear combination of the projections.
b) Show that the natural mapping from $V$ into its bidual ${{V}^{*}}^{*}$ is not surjective.
# Linear mapping/To finite-dimensional/Representation with linear forms/Example/2/Exercise

Describe the linear mapping
    $\varphi \colon K^{4}\longrightarrow K^{3},$
given by the matrix
    ${\begin{pmatrix}3&6&4&5\\5&2&1&3\\8&1&2&7\end{pmatrix}},$
in the sense of fact, using the standard basis and the standard dual basis.
# Dual mapping/Linear subspaces/Exercise

Let
    $\varphi \colon V\longrightarrow W$
be a linear mapping between the finite-dimensional $K$-vector spaces $V$ and $W$, and let
    $\varphi ^{*}\colon {W}^{*}\longrightarrow {V}^{*}$
denote its dual mapping.
a) Show that, for a linear subspace $S\subseteq V$, the relation
    $\varphi ^{*-1}(S^{\perp })=(\varphi (S))^{\perp }\,$
holds.
b) Show that, for a linear subspace $T\subseteq W$, the relation
    $\varphi ^{*}(T^{\perp })=(\varphi ^{-1}(T))^{\perp }\,$
holds.
# Isomorphism/Dual mapping/Linear subspaces/Exercise

Let $\varphi \colon V\rightarrow W$ be an isomorphism between the $K$-vector spaces $V$ and $W$, and let $\varphi ^{*}\colon {W}^{*}\rightarrow {V}^{*}$ denote its dual mapping. Let $U\subseteq V$ be a linear subspace and $T:=\varphi (U)\subseteq W$ the corresponding linear subspace in $W$. Show that, in the dual spaces, the orthogonal spaces $U^{\perp }$ and $T^{\perp }$ correspond to each other.
# Dual mapping/Rank/Exercise

Let
    $\varphi \colon V\longrightarrow W$
be a linear mapping between the finite-dimensional $K$-vector spaces $V$ and $W$, and let
    $\varphi ^{*}\colon {W}^{*}\longrightarrow {V}^{*}$
be the dual mapping. Show
    $\operatorname {rk} \,\varphi =\operatorname {rk} \,\varphi ^{*}\,.$
# Linear mapping/To finite-dimensional/Representation with linear forms/Example/1/Exercise

Describe the linear mapping
    $\varphi \colon K^{3}\longrightarrow K^{2},$
given by the matrix
    ${\begin{pmatrix}6&5&2\\4&8&2\end{pmatrix}},$
in the sense of fact, using the standard basis and the standard dual basis.
# Direct sum/Orthogonal spaces in dual space/Exercise

Let $V$ be a $K$-vector space, together with a direct sum decomposition
    $V=V_{1}\oplus V_{2}\,.$
Show that
    ${V}^{*}=V_{1}^{\perp }\oplus V_{2}^{\perp }\,$
holds, and that the restriction of the dual mapping
    ${V}^{*}\longrightarrow {V_{1}}^{*}$
onto $V_{2}^{\perp }$ is an isomorphism.
# Matrices/Multiplication condition/Interpretation/Exercise

Let $K$ be a field.
a) Let $L$ be an $\ell \times m$-matrix, and let $N$ be an $n\times p$-matrix. Show that
    ${\left\{M\mid M\in \operatorname {Mat} _{m\times n}(K),\,L\circ M\circ N=0\right\}}$
is a linear subspace of $\operatorname {Mat} _{m\times n}(K)$.
b) Let $V$ be an $n$-dimensional and $W$ be an $m$-dimensional $K$-vector space, and let $U\subseteq V$ and $T\subseteq W$ denote linear subspaces. Describe the linear subspace
    ${\left\{\varphi \in \operatorname {Hom} _{K}{\left(V,W\right)}\mid \varphi (U)\subseteq T\right\}}$
using suitable bases and part a).
# Symmetric matrices/By linear forms/Exercise

Describe the space of all symmetric $n\times n$-matrices using linear forms.
# Dual space/Orthogonal space/Dimension formula/Alternative proof/Exercise

Prove fact using fact.
# Linear algebra (Osnabrück 2024-2025)/Linear subspace/Dual space/Orthogonal space/Correspondence/Fact/Factreferencenumber

Lemma 15.6
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/Linear subspaces/Fact/Factreferencenumber

Lemma 13.13
# Dual space/Orthogonal space/Intersection and sum/Exercise

Let $V$ be a $K$-vector space, and let $U_{1},U_{2}\subseteq V$ denote linear subspaces. Show that in the dual space ${V}^{*}$, the equality
    ${\left(U_{1}\cap U_{2}\right)}^{\perp }=U_{1}^{\perp }+U_{2}^{\perp }\,$
holds.
# Dual space/Orthogonal space/Generators/Exercise

Let $U=\langle u_{1},\ldots ,u_{r}\rangle$ be a linear subspace of a $K$-vector space $V$. Show that
    $U^{\perp }={\left\{f\in {V}^{*}\mid f(u_{i})=0{\text{ for all }}i=1,\ldots ,r\right\}}\,$
holds.
# Dual space/Orthogonal space/Linear subspace/Exercise

Let $V$ be a $K$-vector space with its dual space ${V}^{*}$. Show that the orthogonal space $U^{\perp }\subseteq {V}^{*}$ to a linear subspace $U\subseteq V$, and the orthogonal space $F^{\perp }\subseteq V$ to a linear subspace $F\subseteq {V}^{*}$, are indeed linear subspaces.
# (2,5,-1)/Solution space of a linear system/Exercise

Establish a system of linear equations, whose solution space is the line $U={\begin{pmatrix}2\\5\\-1\end{pmatrix}}\mathbb {R}$.
# Fruit seller/Price conditions/Orthogonal space/Exercise

A fruit seller sells apples, pears, and cherries. He cannot remember exactly his purchase prices, but he remembers that he has paid for $5$ kilograms of apples, the same as for $3$ kilograms of pears and one kilogram of cherries together. Moreover, naturally, the old fruit-seller-rule holds that the price of $3$ kilograms of apples equals the price of one kilogram of pears and one kilogram of cherries together. How does the orthogonal space for these price conditions look like? What is the price for one kilogram of apples if the price for one kilogram of cherries is $5$ Euro?
# Linear algebra (Osnabrück 2024-2025)/Part I/News

This is the English version of the material (lecture notes, exercises) of a mathematical course taught at the University of Osnabrück. So this is not an online course, only the material for a real university course. The material is organized in a modular way (e.g., facts, proofs, examples, and exercises are in separate files) so that it can also be used in other courses. The mathematical material is quite standard for a university beginners class, and the course has been taught several times. The English version is a direct translation. Suggestions are welcome.
# Linear algebra (Osnabrück 2024-2025)/Content

Linear algebra is a fundamental course in mathematics. It deals with "linear structures“: linear systems of equations, vectors and their representation, bases and base change, linear mappings and matrices, determinants, eigenvalues and eigenvectors, optimal descriptions of linear mappings, affine mappings, normed vector spaces, lengths and angles, analytic geometry, isometries and symmetries, bilinear forms, more general algebraic structures like equivalence relations, groups, residue class construction, and multilinear algebra.
# Definitions/Role in mathematics/Explanation/Remark

In mathematics, definitions are usually presented as such and get a number so that it is easy to refer to them. The definition contains the description of a situation where only concepts are used that have been defined before. In this situation, a new concept together with a name for it is introduced. This name is printed in a certain font, typically in italic. The new concept can be formulated without the new name; the new name is an abbreviation for the new concept. Quite often, the concepts depend on parameters, like the product set depends on its component sets. The names are often chosen arbitrarily; the meaning of the word within the mathematical context can be understood only via the explicit definition and not via its meaning in everyday life.
# Photography

Photography is the process of making photographs by exposing light over a controlled amount of time to a light-sensitive medium, such as a digital sensor or photographic film. This is done through a device known as a camera. Cameras are essentially light-proof boxes that house the light-sensitive material.

## Materials

In order to properly learn the technical aspects of photography, such as what aperture and shutter speed to use, a single lens reflex (SLR) camera with a fully manual mode of operation is recommended. An entry-level SLR will have all the features needed to build a solid technical foundation.
For handheld photo- and videography, practice steady and stable holding. This enables prolonged exposure times, leading to brighter low-light photos at same light sensitivity, even more so if supported by optical image stabilisation.
For low-light handheld photography, consider using a short self-timer such as two or three seconds to avoid causing a shake during exposure by pressing the shutter button. This is especially necessary if the camera supports zero shutter lag/delay, but the firmware can not optimize low light photographs from multiple still images.
If one does approach learning photography with a film camera, it is a good idea to spend time learning how to develop the negatives and make prints, which allows the photographer employ further techniques such as push processing and an appreciation of the chemical processes involved in the craft of creating analog photographic images. Digital cameras provide instant feedback, which allows one to learn from their photograph on the spot and perhaps make another attempt.
Photography Stream

## Courses / projects

* Concert photography
* Composition
* Digital Asset Management (DAM)
* Digital Darkroom
* Digital photography
* Digital Workflow
* Film photography
* Legal Issues in Photography
* Metadata
* Museum photography
* Photojournalism
* Street Photography
* Apply design elements in photography
* The art of photography
# Matrix/Column rank/Row rank/Introduction/Section
## Definition

Let $K$ be a field, and let $M$ denote an $m\times n$-matrix over $K$. Then the dimension of the linear subspace of $K^{m}$, generated by the columns, is called the column rank of the matrix, written
    $\operatorname {rk} \,M.$

## Lemma

Let $K$ denote a field, and let $V$ and $W$ denote $K$-vector spaces of dimensions $n$ and $m$. Let
    $\varphi \colon V\longrightarrow W$
be a linear mapping which is described by the matrix $M\in \operatorname {Mat} _{m\times n}(K)$, with respect to bases of the spaces. Then
    $\operatorname {rk} \,\varphi =\operatorname {rk} \,M\,$

### Proof

$\Box$
To formulate the next statement, we introduce row rank of an $m\times n$-matrix to be the dimension of the linear subspace of $K^{n}$ generated by the rows.

## Lemma

Let $K$ be a field, and let $M$ denote an $m\times n$-matrix over $K$. Then the column rank coincides with the row rank. The rank equals the number $r$ from

### Proof

In an elementary row manipulation, the linear subspace generated by the rows is not changed, therefore the row rank is not changed. The row rank of $M$ equals the row rank of the matrix in echelon form obtained in fact. This matrix has row rank $r$, since the first $r$ rows are linearly independent, and beside this, there are only zero rows. It has also column rank $r$, since the $r$ columns, where there is a new step, are linearly independent, and the other columns are linear combinations of these $r$ columns. By exercise, the column rank is preserved by elementary row manipulations.
$\Box$
Both ranks coincide, so we only talk about the rank of a matrix.

## Corollary

Let $K$ be a field, and let $M$ denote an $n\times n$-matrix
$K$
* $M$ is invertible.
* The rank of $M$ is $n$.
* The rows of $M$ are linearly independent.
* The columns of $M$ are linearly independent.

### Proof

$\Box$
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Matrix to basis/Several properties/Fact/Factreferencenumber

Lemma 12.5
# Linear algebra (Osnabrück 2024-2025)/Matrix/Row rank and column rank/Fact/Factreferencenumber

Lemma 12.15
# Linear algebra (Osnabrück 2024-2025)/Elementary matrices/Invertible/Explicit/Exercise/Exercisereferencenumber

Exercise 12.1
# Linear algebra (Osnabrück 2024-2025)/Endomorphismus/Finite dimensional/Change of basis/Fact/Factreferencenumber

Corollary 11.12
# MDLD/coordinate (vs)

Start=coordinate (vs) Target=Vector space/Basis and isomorphic to K^n/Remark
# MDLD/standard vector (vs)

Start=standard vector (vs) Target=Vector space/Examples/Introduction/Section
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Image and preimage/Linear subspaces/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 11.2
# MDLD/finite dimension (vs)

Start=finite dimension (vs) Target=Vector space/Finitely generated/Dimension/Definition
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Kernel/Injectivity/Fact/Factreferencenumber

Lemma 11.4
# MDLD/coordinate mappings

Start=coordinate mappings Target=Vector space/Basis and isomorphic to K^n/Remark
# Linear algebra (Osnabrück 2024-2025)/Matrix multiplication/Composition/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 10.20
# Linear algebra (Osnabrück 2024-2025)/Linear mapping/Finite dimensional/Change of basis/Fact/Factreferencenumber

Lemma 11.11
# MDLD/elementary row manipulation

Start=elementary row manipulation Target=Matrix/Elementary row operation/Definition
# Linear algebra (Osnabrück 2024-2025)/Matrix/Staircase form by elementary manipulations/Fact/Factreferencenumber

Theorem 12.9
# Vector space/Basis/Subfamily/Projection/Example

Let $V$ be a finite-dimensional $K$-vector space and $v_{i}$, $i\in I$, a basis of $V$. For a subset $J\subseteq I$, set
    $V_{J}=\langle v_{i},\,i\in J\rangle \,,$
the linear subspace corresponding to $J$. Moreover, let
    $p_{J}\colon V\longrightarrow V,\sum _{i\in I}a_{i}v_{i}\longmapsto \sum _{i\in J}a_{i}v_{i},$
the corresponding projection. The image of this projection is $V_{J}$. On $V_{J}$, this mapping is the identity, one can also consider this mapping as
    $p_{J}\colon V\longrightarrow V_{J}.$
The kernel of this mapping is
    $\operatorname {kern} p_{J}=\langle v_{i},\,i\notin J\rangle \,.$
# MDLD/projection (direct sum)

Start=projection (direct sum) Target=Vector space/Projection/Introduction/Section
# Linear algebra (Osnabrück 2024-2025)/Vector space/Basis/Subfamily/Projection/Example/Examplereferencenumber

Example 13.2
# Projection/Linear subspace/Idempotent/Fact

Let $K$ be a field and $V$ a $K$-vector space.
direct sum decomposition
    $V=U\oplus U'\,,$
the projection onto $U$ is a projection in the sense of Definition. Such a projection
    $\varphi \colon V\longrightarrow V$
gives a decomposition
    $V=\operatorname {kern} \varphi \oplus \operatorname {Im} \varphi \,,$
$\varphi$
$\operatorname {Im} \varphi$
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/Vector space/Exercise/Exercisereferencenumber

Exercise 13.8
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/K to W/Example/Exercise/Exercisereferencenumber

Exercise 13.10
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/Functorial properties/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 13.15
# MDLD/canonical projections (direct sum)

Start=canonical projections (direct sum) See=projection (direct sum) MDLD/projection (direct sum) Target=Vector space/Projection/Introduction/Section
# Linear algebra (Osnabrück 2024-2025)/Direct sum/Linear mapping/Fact/Factreferencenumber

Lemma 13.9
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/Bases/Matrices/Fact/Factreferencenumber

Theorem 13.11
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/Dimension/Fact/Factreferencenumber

Corollary 13.12
# Linear algebra (Osnabrück 2024-2025)/Homomorphism space/Direct sum decomposition/Fact/Factreferencenumber

Lemma 13.10
# Linear algebra (Osnabrück 2024-2025)/Linear independence/Test with linear forms/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 14.7
# Linear algebra (Osnabrück 2024-2025)/Trace/V over K/Independent of basis/Exercise/Exercisereferencenumber

Exercise 14.15
# Linear algebra (Osnabrück 2024-2025)/Dual space/Orthogonal space/Linear subspace/Exercise/Exercisereferencenumber

Exercise 15.4
# Linear algebra (Osnabrück 2024-2025)/Dual space/Orthogonal space/Generators/Exercise/Exercisereferencenumber

Exercise 15.5
# Vector space/Bidual/Natural mapping/Fact

Let $K$ be a field, and let $V$ be a $K$-vector space.
injective linear mapping
    $\Psi \colon V\longrightarrow {({V}^{*})}^{*},v\longmapsto {\left(f\mapsto f(v)\right)}.$
If $V$ has finite dimension, then $\Psi$ is an
# Linear algebra (Osnabrück 2024-2025)/Linear form/Vector/Composition/Remark/Remarkreferencenumber

Remark 14.4
# Linear algebra (Osnabrück 2024-2025)/Multilinear mapping/Distributivity law/Fact/Proof/Exercise/Exercisereferencenumber

Exercise 16.17
# Linear algebra (Osnabrück 2024-2025)/Multilinear mapping/Distributivity law/Fact/Factreferencenumber

Lemma 16.6
# Linear algebra (Osnabrück 2024-2025)/Determinant/Recursive/Multilinear/Scalar/Exercise/Exercisereferencenumber

Exercise 16.22
# Linear algebra (Osnabrück 2024-2025)/Alternating mapping/Swap property/Fact/Factreferencenumber

Lemma 16.8
# Linear algebra (Osnabrück 2024-2025)/Quadratic matrix/Rank/Invertible/Linearly independent/Fact/Factreferencenumber

Corollary 12.16
# Linear algebra (Osnabrück 2024-2025)/Determinant/Recursively/Multilinear/Fact/Factreferencenumber

Theorem 16.9
# Linear algebra (Osnabrück 2024-2025)/Determinant/Recursively/Alternating/Swap property/Fact/Factreferencenumber

Theorem 16.10
# Linear algebra (Osnabrück 2024-2025)/Determinant/Field/Upper triangular matrix/Fact/Factreferencenumber

Lemma 16.4
# Linear algebra (Osnabrück 2024-2025)/Determinant function/Behavior under row operations/Fact/Factreferencenumber

Lemma 17 2.
# Linear algebra (Osnabrück 2024-2025)/Determinant/Recursively/Alternating/Fact/Factreferencenumber

Theorem 16.10
# Linear algebra (Osnabrück 2024-2025)/Determinant/Zero, linear dependent and rank property/Fact/Factreferencenumber

Theorem 16.11
# Linear algebra (Osnabrück 2024-2025)/Determinant function/Uniquely determined/Fact/Factreferencenumber

Theorem 17.3
# Determinant/Multiplication theorem/Fact/Proof

We fix the matrix $B$.
Suppose first that $\det B=0$. Then, due to fact the matrix $B$ is not invertible and therefore, also $A\circ B$ is not invertible. Hence, $\det A\circ B=0$.
Suppose now that $B$ is invertible. In this case, we consider the well-defined mapping
    $\delta \colon \operatorname {Mat} _{n}(K)\longrightarrow K,A\longmapsto (\det A\circ B)(\det B)^{-1}.$
We want to show that this mapping equals the mapping $A\mapsto \det A$, by showing that it fulfills all the properties which, according to fact, characterize the determinant. If $z_{1},\ldots ,z_{n}$ denote the rows of $A$, then $\delta (A)$ is computed by applying the determinant to the rows $z_{1}B,\ldots ,z_{n}B$, and then by multiplying with $(\det B)^{-1}$. Hence the multilinearity and the alternating property follows from exercise. If we start with $A=E_{n}$, then $A\circ B=B$ and thus
    $\delta (E_{n})=(\det B)\cdot (\det B)^{-1}=1\,.$
# Linear algebra (Osnabrück 2024-2025)/Multilinear mapping/Constant factor/Alternating/Linear product mapping first/Exercise/Exercisereferencenumber

Exercise 16.29