OpenLLM-France/wikimedia huggingface dataset:
# Disjoint sets/Moving/Bijection/Exercise

Let $M$ and $N$ be disjoint sets, and $x\in M$ Show that also $M\setminus \{x\}$ and $N\cup \{x\}$ are disjoint, and that
    $M\cup N={\left(M\setminus \{x\}\right)}\cup {\left(N\cup \{x\}\right)}\,$
holds.
# Space/Plane equation/Example for intersection/2/Exercise

We consider the two sets
    $E={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in \mathbb {R} ^{3}\mid -3x+2y-6z=0\right\}}\,$
and
    $F={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in \mathbb {R} ^{3}\mid 7x-5y-4z=0\right\}}\,.$
Find a description of the intersection
    $G:=E\cap F={\left\{{\begin{pmatrix}x\\y\\z\end{pmatrix}}\in \mathbb {R} ^{3}\mid -3x+2y-6z=0{\text{ and }}7x-5y-4z=0\right\}}\,,$
similar to example.
# Product set/Distributivity law/Exercise

Let $A$ and $B$ denote disjoint sets, and let $C$ be another set. Show the identity
    $C\times (A\uplus B)=(C\times A)\uplus (C\times B)\,.$
# Product set/Binomial formula/Exercise

Let $A$ and $B$ denote disjoint sets. Show the equality
    $(A\uplus B)\times (A\uplus B)=(A\times A)\uplus (A\times B)\uplus (B\times A)\uplus (B\times B)\,.$
# Several subsets im R^2/Sketch/2/Exercise

Sketch the following subsets in $\mathbb {R} ^{2}$.
* ${\left\{(x,y)\mid \vert {2x}\vert =5{\text{ and }}\vert {y}\vert \geq 3\right\}}$,
* ${\left\{(x,y)\mid -3x\geq 2y{\text{ and }}4x\leq -5y\right\}}$,
* ${\left\{(x,y)\mid y^{2}-y+1\leq 4\right\}}$,
* ${\left\{(x,y)\mid xy=2{\text{ or }}x^{2}+y^{2}=1\right\}}$.
# Set theory/Aristotelian Syllogisms/1/Exercise

Prove the following (set-theoretical versions of) syllogisms of Aristotle. Let $A,B,C$ denote sets.
* Modus Barbara: $B\subseteq A$ and $C\subseteq B$ imply $C\subseteq A$.
* Modus Celarent: $B\cap A=\emptyset$ and $C\subseteq B$ imply $C\cap A=\emptyset$.
* Modus Darii: $B\subseteq A$ and $C\cap B\neq \emptyset$ imply $C\cap A\neq \emptyset$.
* Modus Ferio: $B\cap A=\emptyset$ and $C\cap B\neq \emptyset$ imply $C\not \subseteq A$.
* Modus Baroco: $B\subseteq A$ and $B\not \subseteq C$ imply $A\not \subseteq C$.
# Addition/Multiplication/Power/N/Injectivity/Exercise

Is the mapping
    $\varphi \colon \mathbb {N} _{+}\times \mathbb {N} _{+}\longrightarrow \mathbb {N} _{+}\times \mathbb {N} _{+}\times \mathbb {N} _{+},(a,b)\longmapsto (a+b,ab,a^{b}),$
injective, or not?
# Mapping/Composition/Surjective/Fact/Proof/Exercise

Let $L,M,N$ be sets, let $F\colon L\rightarrow M$ and $G\colon M\rightarrow N$ denote surjective mappings. Show that the composition $G\circ F$ is also surjective.
# Mapping/Composition/Injective/Fact/Proof/Exercise

Let $L,M,N$ be sets, let $F\colon L\rightarrow M$ and $G\colon M\rightarrow N$ denote injective mappings. Show that the composition $G\circ F$ is also injective.
# Rational numbers/Squaring/Exercise

Is the mapping
    $\mathbb {Q} _{\geq 0}\longrightarrow \mathbb {Q} _{\geq 0},x\longmapsto x^{2},$
injective? Is it surjective?
# Power set/Indicator function/Bijection/Exercise

Let $G$ be a set. Define a bijection between
    ${\mathfrak {P}}\,(G)\,\,{\text{ and }}\,\,\operatorname {Map} \,{\left(G,\{0,1\}\right)}.$
# Set/Disjoint union/Bijection of the power sets/2/Exercise

Let $G$ be a set, which is given as a disjoint union
    $G=A\uplus B\,.$
Define a bijection between the power set ${\mathfrak {P}}\,(G)$ and the product set ${\mathfrak {P}}\,(A)\times {\mathfrak {P}}\,(B)$.
# Mapping/Take preimage/Complete assignment/Exercise

Let $L$ and $M$ denote sets. We consider the mapping
    $\Psi \colon \operatorname {Map} \,{\left(L,M\right)}\longrightarrow \operatorname {Map} \,{\left({\mathfrak {P}}\,(M),{\mathfrak {P}}\,(L)\right)},f\longmapsto f^{-1},$
which assigns to a mapping the mapping given by taking preimages.
a) Show that $\Psi$ is injective.
b) Suppose $L\neq \emptyset$. Show that $\Psi$ is not surjective.
# Sets/M,N,L/Map(MxN,L) and Map(M,Map(N,L))/Bijection/Exercise

Let $M,N,L$ denote sets. Define a bijection between
    $\operatorname {Map} \,{\left(M\times N,L\right)}\,\,{\text{ and }}\,\,\operatorname {Map} \,{\left(M,\operatorname {Map} \,{\left(N,L\right)}\right)}.$
# Mapping/Injective and preimage surjective/Exercise

Let $L$ and $M$ denote sets, and let
    $F\colon L\longrightarrow M$
be a mapping. Show that $F$ is injective if and only if taking preimages
    ${\mathfrak {P}}\,(M)\longrightarrow {\mathfrak {P}}\,(L),T\longmapsto F^{-1}(T),$
is surjective.
# Mapping/Surjective and preimage injective/Exercise

Let $L$ and $M$ denote sets, and let
    $F\colon L\longrightarrow M$
be a mapping. Show that $F$ is surjective if and only if taking preimages
    ${\mathfrak {P}}\,(M)\longrightarrow {\mathfrak {P}}\,(L),T\longmapsto F^{-1}(T),$
is injective.
# Mapping/Take image/Exercise

Let $F\colon L\rightarrow M$ be a mapping. Show that taking images
    ${\mathfrak {P}}\,(L)\longrightarrow {\mathfrak {P}}\,(M),S\longmapsto F(S),$
satisfies the following properties (for arbitrary subsets $S,S_{1},S_{2}\subseteq L$)
* $F(S_{1}\cap S_{2})\subseteq F(S_{1})\cap F(S_{2})$,
* $F(S_{1}\cup S_{2})=F(S_{1})\cup F(S_{2})$,
* $F(L\setminus S)\supseteq F(L)\setminus F(S)$.
Show by examples that the inclusions in (1) and (3) might be strict.
# Graph (mapping)/R and R^2/Imagination/Exercise

How can we visualize the graph of a mapping
    $\varphi \colon \mathbb {R} ^{2}\longrightarrow \mathbb {R} ,$
and the graph of a mapping
    $\varphi \colon \mathbb {R} \longrightarrow \mathbb {R} ^{2}?$
# Integers/Difference/Structural properties/Exercise

Consider the integers $\mathbb {Z}$ together with the difference as a binary operation, that is, the mapping
    $\mathbb {Z} \times \mathbb {Z} \longrightarrow \mathbb {Z} ,(a,b)\longmapsto a-b.$
Does there exist a neutral element for this operation? Is this operation associative, commutative, does there exist for every element an inverse element?
# Real nonnegative numbers/Minimum/Structural properties/Exercise

Study the binary operation
    $\mathbb {R} _{\geq 0}\times \mathbb {R} _{\geq 0}\longrightarrow \mathbb {R} _{\geq 0},(x,y)\longmapsto \operatorname {min} \,(x,y),$
with respect to associativity, commutativity, existence of a neutral element, existence of inverse elements.
# Operation table/Mappings on set with 2 elements/Exercise

Let $M$ be the set of all mappings from the set $\{0,1\}$ to itself, that is
    $M={\left\{F:\{0,1\}\rightarrow \{0,1\}\mid F{\text{ mapping}}\right\}}\,.$
Denote the elements of $M$ with certain symbols, and establish a value table for the binary operation on $M$ given by the composition of mappings.
# Mappings/Bijective mappings/Group/Exercise

Let $S$ be a set and define
    $G={\left\{F:S\rightarrow S\mid F{\text{ bijective}}\right\}}\,.$
Show that $G$ is, with the composition of mappings as operation, a group.
# Group/Inverse of xy/Exercise

Let $G$ be a group, and $x,y\in G$. Express the inverse of $xy$ with the inverses of $x$ and $y$.
# Group/3 elements/Exercise

Construct a group with three elements.
# Commutative ring/Sum and product of polynomial terms/Exercise

Let $R$ be a commutative ring, and $f,a_{i},b_{j}\in R$. Show the following equations:
    $[[:Template:Polynomialoneadditionformula]]$
and
    $[[:Template:Polynomialonemultiplicationformula]].$
# Graph (mapping)/R/Addition and multiplication/Exercise

Sketch the graph of the real addition
    $+\colon \mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ,(x,y)\longmapsto x+y,$
and the graph of the real multiplication
    $\cdot \colon \mathbb {R} \times \mathbb {R} \longrightarrow \mathbb {R} ,(x,y)\longmapsto x\cdot y.$
# Complex numbers/General binomial formula/Exercise

Compute
    $(x+{\mathrm {i} }y)^{n}.$
# Field/Zero set/Nearly a field/Exercise

Show that the set $\{0\}$ satisfies all axioms for a field, except that $0=1$ holds.
# Field/Qi/Exercise

We consider the set
    $K=\mathbb {Q} \times \mathbb {Q} ={\left\{(a,b)\mid a,b\in \mathbb {Q} \right\}}\,$
with the special elements
    $0=(0,0){\text{ and }}1=(1,0),$
the addition
    $(a,b)+(c,d):=(a+c,b+d)\,$
and the multiplication
    $(a,b)\cdot (c,d):=(ac-bd,ad+bc)\,.$
Show that $K$ is, with these operations, a field.
# Binary operation/Table/4/2/Exercise

On the set
    $M=\{a,b,c,d\}\,,$
we consider the binary operation $\star$ given by the table
* Compute
    $b\star (a\star (d\star a)).$
* Does there exist a neutral element for $\star$?
# Intersection/Power set/Operation/Exercise

Let $G$ be a set, and let $M={\mathfrak {P}}\,(G)$ be the corresponding power set. Consider the intersection of subsets of $G$ as a binary operation on $M$. Is this operation commutative, associative, does there exist a neutral element?
# Binary operation/Associative/5 factors/1/Exercise

Let $M$ be a set, with an associative binary operation $\star$ defined on it. Show that
    $(a\star b)\star (c\star (d\star e))=a\star ((b\star (c\star d))\star e)\,$
holds for arbitrary $a,b,c,d,e\in M$.
# Field/Binomi/Explained/Fact/Proof/Exercise

Prove the general binomial formula, that is, the formula
    $(a+b)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}a^{k}b^{n-k}$
for $n\in \mathbb {N}$ and arbitrary elements $a,b\in K$ in a field $K$.
# Dimensional Balancing

Dimensional balancing is an extension of dimensional analysis. In dimensional balancing the dimensional profile on the right-hand side of the equation is always equal to the dimensional profile on the left-hand side of the equation.
The notation used in dimensional balancing is as follows;
* each entity is assigned a dimensional power,
** $D_{\;Physical\;Concept}^{Dimensional\;Exponent/Power}$
* all equation the dimensional power of the left hand side of the equation must equal the dimensional power of the right hand side
For example, based on the below table energy has a dimension of three which is equal to mass of dimension three times speed of light squared which has a dimensions of zero or 3 = 3 + (0*2). The left hand side of the equation must equal the right hand side of the equation.
* Energy = kg * $c^{2}$
* $D_{E}^{3}$ = $D_{kg}^{3}$ * $D_{c}^{0*2}$
| Unit | Dimension | Einstein Units |
| kg = mass | $D_{kg}^{3}$ | ${\frac {c}{AB}}$ |
| E = energy | $D_{E}^{3}$ | ${\frac {c^{3}}{AB}}$ |
| F = force | $D_{F}^{2}$ | ${\frac {c^{2}}{B}}$ |
| c = speed of light | $D_{c}^{0}$ | c |
| m = distance | $D_{m}^{1}$ | ${\frac {c}{A}}$ |
| s = time | $D_{s}^{1}$ | ${\frac {1}{A}}$ |
| h = Planck Constant | $D_{h}^{4}$ | ${\frac {c^{3}}{A^{2}B}}$ |
| Q = charge | $D_{Q}^{2}$ | ${\frac {c^{2}}{AB^{0.5}}}$ |
| A = 1/time | $D_{A}^{-1}$ | $A$ |
| B = m/kg | $D_{B}^{-2}$ | $B$ |
| Electric Field ${\mathbf {E} }$ | ${\frac {D_{F}^{2}}{D_{q}^{2}}}=D_{\mathbf {E} }^{0}$ | ${\frac {A}{\pm B^{0.5}}}$ |
| Magnetic Field ${\mathbf {B} }$ | ${\frac {D_{F}^{2}}{D_{q}^{2}\;D_{\mathbf {v} }^{0}}}=D_{\mathbf {B} }^{0}$ | ${\frac {A}{\pm cB^{0.5}}}$ |
| Gravitational Field ${\mathbf {g} }$ | ${\frac {D_{F}^{2}}{D_{m}^{3}}}=D_{\mathbf {g} }^{-1}$ | $Ac$ |
# Linear system/2x+3y is 7 and 5x+4y is 3/Exercise

Solve the linear system
    $2x+3y=7{\text{ and }}5x+4y=3.$
# Linear equation/(2+5i)z is (3-7i)/Modulus/Exercise

Solve the linear equation
    $(2+5{\mathrm {i} })z=(3-7{\mathrm {i} })\,$
over $\mathbb {C}$, and compute the modulus of the solution.
# Linear system/Over Z mod 7/1/Exercise

Solve the linear system
    ${\begin{pmatrix}3&4\\2&6\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}5\\1\end{pmatrix}}\,$
over the field $K=\{0,1,2,3,4,5,6\}$ from example.
# Linear system/3x is 5/Several fields/1/Exercise

Solve the linear equation
    $3x=5\,$
for the following fields $K$:
a) $K=\mathbb {Q}$,
b) $K=\mathbb {R}$,
c) $K=\{0,1\}$, the field with two elements from example,
d) $K=\{0,1,2,3,4,5,6\}$, the field with seven elements from example.
# Matrix multiplication/Complex matrices/Exercise

Compute the product of matrices
    ${\begin{pmatrix}2+{\mathrm {i} }&1-{\frac {1}{2}}{\mathrm {i} }&4{\mathrm {i} }\\-5+7{\mathrm {i} }&{\sqrt {2}}+{\mathrm {i} }&0\end{pmatrix}}{\begin{pmatrix}-5+4{\mathrm {i} }&3-2{\mathrm {i} }\\{\sqrt {2}}-{\mathrm {i} }&e+\pi {\mathrm {i} }\\1&-{\mathrm {i} }\end{pmatrix}}{\begin{pmatrix}1+{\mathrm {i} }\\2-3{\mathrm {i} }\end{pmatrix}},$
according to the two possible parentheses.
# Linear system/Over Q sqrt(3)/Exercise

Solve the following linear system over the field $K=\mathbb {Q} [{\sqrt {3}}]$:
    ${\begin{pmatrix}2+{\sqrt {3}}&-{\sqrt {3}}\\{\frac {1}{2}}&-2-3{\sqrt {3}}\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}={\begin{pmatrix}1-{\sqrt {3}}\\4-2{\sqrt {3}}\end{pmatrix}}\,.$
# Linear system/Substitution method/1/Exercise

Solve the linear system
    ${\begin{matrix}4x&+7y&+3z&=&4\\9x&+13y&+6z&=&9\\5x&y&z&=&2\,\end{matrix}}$
with the substitution method.
# Linear system/Equating method/1/Exercise

Solve the linear system
    ${\begin{matrix}5x&+6y&+2z&=&6\\8x&+9y&+11z&=&5\\7x&y&z&=&8\,\end{matrix}}$
with the equating method.
# Linear system/Q/Equivalent with small numbers/Exercise

Show that, for every linear system over $\mathbb {Q}$, there exists an equivalent system with the property that the modulus of all coefficients is smaller than $1$.
# Linear system/Simultaneous/1/Exercise

Solve the linear systems
    ${\begin{pmatrix}7&4\\1&2\end{pmatrix}}{\begin{pmatrix}x_{1}\\y_{1}\end{pmatrix}}={\begin{pmatrix}4\\9\end{pmatrix}},\,{\begin{pmatrix}7&4\\1&2\end{pmatrix}}{\begin{pmatrix}x_{2}\\y_{2}\end{pmatrix}}={\begin{pmatrix}6\\5\end{pmatrix}},\,{\begin{pmatrix}7&4\\1&2\end{pmatrix}}{\begin{pmatrix}x_{3}\\y_{3}\end{pmatrix}}={\begin{pmatrix}-2\\5\end{pmatrix}}$
simultaneously.
# Linear system/Superposition principle/Fact/Proof/Exercise

Prove the superposition principle for systems of linear equations.
# Linear system/Parameter/Size of solution/Exercise

We consider the linear system
    ${\begin{matrix}2x&-ay&&=&-2\\ax&&+3z&=&3\\-{\frac {1}{3}}x&+y&+z&=&2\end{matrix}}$
over the real numbers, depending on the parameter $a\in \mathbb {R}$. For which $a$ does the system of equations have no solution, one solution, or infinitely many solutions?
# Symmetric 2x2-matrices/Vector space/Exercise

Show that the set of all "symmetric“ $2\times 2$-matrices over a field $K$, that is, matrices of the form
    ${\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}$
satisfying the condition
    $a_{12}=a_{21}\,,$
is, with componentwise addition and componentwise scalar multiplication, a $K$-vector space.
# Vector space/One axiom missing/Identity/Example/Exercise

Give an example of a field $K$, a commutative group $(V,+,0)$, and a mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms, with the exception of
    $(5)\,\,\,1u=u.$
# Vector space/One axiom missing/Associativity/Example/Exercise

Give an example of a field $K$, a commutative group $(V,+,0)$, and a mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms, with the exception of
    $(6)\,\,\,r(su)=(rs)u.$
# Vector space/One axiom missing/Distributivity in the vector space/Example/Exercise

Give an example of a field $K$, a commutative group $(V,+,0)$, and a mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms, with the exception of
    $(7)\,\,\,r(u+v)=ru+rv.$
# Vector space/One axiom missing/Distributivity in the field/Example/Exercise

Give an example of a field $K$, a commutative group $(V,+,0)$, and a mapping
    $K\times V\longrightarrow V,(s,v)\longmapsto sv,$
such that this structure fulfills all vector space axioms, with the exception of
    $(8)\,\,\,(r+s)u=ru+su.$
# Vector space/Surjective mapping with structure/Vector space structure/Exercise

Let $K$ be a field, $V$ a $K$-vector space, and $M$ a set with a binary operation
    $+\colon M\times M\longrightarrow M,$
and a mapping
    $\cdot \colon K\times M\longrightarrow M.$
Let
    $\varphi \colon V\longrightarrow M$
be a surjective mapping satisfying
    $\varphi (x+y)=\varphi (x)+\varphi (y)\,\,{\text{ and }}\,\,\varphi (sx)=s\varphi (x)$
for all $x,y\in V$ and $s\in K$. Show that $M$ is a $K$-vector space.
# Vector space/Linear subspace/I to K/J in I/Exercise

let $K$ be a field and let $J\subseteq I$ denote two index sets. Show that $K^{J}=\operatorname {Map} \,{\left(J,K\right)}$ is, in a natural way, a linear subspace of $K^{I}$.
# Vector space/Set of sequences in ordered K/Cauchy-sequences as linear subspace/Exercise

Let $K$ denote an ordered field, and set
    $C={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{Cauchy sequence in }}K\right\}}\,.$
Show that $C$ is a linear subspace of the space of sequences
    $F={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid {\text{sequence in }}K\right\}}\,.$
# Linear combination/R/(1,0,0)/By (1,-2,5), (4,0,3) and (2,1,1)/Exercise

Express, in $\mathbb {R} ^{3}$, the vector
    $(1,0,0)$
as a linear combination of the vectors
    $(1,-2,5),(4,0,3){\text{ and }}(2,1,1).$
# Decide linear independence/Exercise

Decide whether the following vectors are linearly independent.
* $(-1,1,-1)$, $(0,6,4)$, $(1,2,3)$, in the $\mathbb {R}$-vector space $\mathbb {R} ^{3}$.
* $1+{\mathrm {i} }$, $1+2{\mathrm {i} }$ in the $\mathbb {R}$-vector space $\mathbb {C}$.
* $1+{\mathrm {i} }$, $1+2{\mathrm {i} }$ in the $\mathbb {C}$-vector space $\mathbb {C}$.
* $1$, ${\sqrt {3}}$ in the $\mathbb {Q}$-vector space $\mathbb {R}$.
# Linearly independent/Basis in linear subspace/Exercise

Let $V$ be a $K$-vector space, and let $v_{1},\ldots ,v_{n}$ be a family of vectors in $V$. Show that the family is linearly independent if and only if there exists a linear subspace $U\subseteq V$ such that the family is a basis of $U$.
# Basis of linear subspace/1/Exercise

Determine a basis for the linear subspace
    $U={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x=z\right\}}\subset \mathbb {R} ^{3}\,.$
# Linear subspace/Q^n/Integer basis/Exercise

Let $U\subseteq \mathbb {Q} ^{n}$ be a linear subspace. Show that $U$ has a basis consisting of vectors, such that all their entries are integer numbers.
# Vector space/Countable basis/Flags/Exchange/Exercise

Let $V$ be a $K$-vector space, and let $v_{n}$, $n\in \mathbb {N} _{+}$, be a basis of $V$. Let $u_{n}$, $n\in \mathbb {N} _{+}$, be another family of vectors in $V$. Suppose that, for every $n\in \mathbb {N}$, the equality
    $\langle v_{1},\ldots ,v_{n}\rangle =\langle u_{1},\ldots ,u_{n}\rangle \,$
holds. Show that also $u_{n}$, $n\in \mathbb {N} _{+}$, is a basis of $V$.
# Polynomial ring/Basis/Sum polynomials/Exercise

Let $\mathbb {R} [X]$ be the polynomial ring over $\mathbb {R}$. For $n\in \mathbb {N}$, set
    $F_{n}=1+2X+3X^{2}+\cdots +(n+1)X^{n}\,.$
Show that $F_{n}$, $n\in \mathbb {N}$, is a basis of $\mathbb {R} [X]$.
# Vector space/Characterizations of basis/Maximal/Minimal/Infinite-dimensional/Exercise

Formulate and prove fact for an arbitrary (not necessarily finite) family of vectors $v_{i}$, $i\in I$.
# Prime numbers/Logarithms/Linearly independent/Tip/Exercise

We consider the real numbers $\mathbb {R}$ as a $\mathbb {Q}$-vector space. Show that the set of real numbers $\ln p$, where $p$ runs through the set of all prime numbers, is
linearly independent. Tip: Use that every positive natural number has a unique representation as a product of prime numbers.
# Vector space/Theorem of Hamel/Fact

Every vector space
# Linear subspaces/Intersection/(2 1 7), (4 -2 9) and (3 1 0), (5 2 -4)/Exercise

In $\mathbb {R} ^{3}$, let the two linear subspaces
    $U={\left\{s{\begin{pmatrix}2\\1\\7\end{pmatrix}}+t{\begin{pmatrix}4\\-2\\9\end{pmatrix}}\mid s,t\in \mathbb {R} \right\}}\,$
and
    $V={\left\{p{\begin{pmatrix}3\\1\\0\end{pmatrix}}+q{\begin{pmatrix}5\\2\\-4\end{pmatrix}}\mid p,q\in \mathbb {R} \right\}}\,$
be given. Determine a basis for $U\cap V$.
# K^n/Multiplication in coordinate system/Not compatible/Exercise

Let $v_{1},\ldots ,v_{n}$ be a basis of $K^{n}$, and let
    $\psi _{\mathfrak {v}}\colon K^{n}\longrightarrow K^{n},{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n},$
be the corresponding bijective mapping in the sense of remark. Show that this mapping is, in general, not compatible with componentwise multiplication in $K^{n}$.
# Space of matrices/Dimension/Exercise

Determine the dimension of the space of all $m\times n$-matrices.
# Linear system/Dimension/1/Exercise

Determine the dimension of the solution space of the linear system
    $4x-3y+7z+5u-v=0\,$
and
    $y+6z-10u+3v=0\,$
in the variables $x,y,z,u,v$.
# Diagonal matrices/Linear subspace and dimension/Exercise

Let $K$ be a field, and $n\in \mathbb {N}$. Show that the set of all diagonal matrices is a linear subspace in the space of all $n\times n$-matrices over $K$, and determine its dimension.
# Symmetric matrices/Linear subspace/Dimension/Exercise

Show that the set of all symmetric $n\times n$-matrices is a linear subspace in the space of all $n\times n$-matrices, and determine its dimension.
# Upper triangular matrices/Linear subspace and dimension/Exercise

Let $K$ be a field, and $n\in \mathbb {N}$. Show that the set of all upper triangular matrices is a linear subspace in the space of all $n\times n$-matrices over $K$, and determine its dimension.
# Vector space/Finite dimension/Linear subspace of full dimension/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space of finite dimension. Let $U\subseteq V$ denote a linear subspace with $\dim _{\left(U\right)}=\dim _{\left(V\right)}$. Show that $U=V$ holds.
# Parametrized vectors/abc/Cyclically swapped/Dimensions/Exercise

Let $a,b,c\in \mathbb {R}$ be real numbers. We consider the three vectors
    ${\begin{pmatrix}a\\b\\c\end{pmatrix}},\,{\begin{pmatrix}c\\a\\b\end{pmatrix}},\,{\begin{pmatrix}b\\c\\a\end{pmatrix}}\in \mathbb {R} ^{3}\,.$
Give examples for $a,b,c$ such that the linear subspace generated by these vectors has dimension $0,1,2,3$.
# Linear subspaces/Sum of the dimensions larger/Intersection/Exercise

Let $W$ be an $n$-dimensional $K$-vector space ($K$ a field), and let $U,V\subseteq W$ be linear subspaces of dimension $\dim _{K}{\left(U\right)}=r$ and $\dim _{K}{\left(V\right)}=s$. Suppose that $r+s>n$ holds. Show that $U\cap V\neq 0$.
# Infinite basis/Finite basis/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Show that $V$ can not have a finite basis and an infinite basis.
# Base change/Same basis/Identity matrix/Exercise

Determine the transformation matrix $M_{\mathfrak {v}}^{\mathfrak {v}}$ for the identical base change of ${\mathfrak {v}}$ to ${\mathfrak {v}}$.
# Base change/Exchange lemma/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space, with a basis ${\mathfrak {v}}=v_{1},\ldots ,v_{n}$. Let $w\in V$ be a vector with a representation
    $w=\sum _{i=1}^{n}s_{i}v_{i}\,,$
where $s_{k}\neq 0$ for a certain $k$. Let
    ${\mathfrak {w}}=v_{1},\ldots ,v_{k-1},w,v_{k+1},\ldots ,v_{n}\,.$
Determine the transformation matrices $M_{\mathfrak {w}}^{\mathfrak {v}}$ and $M_{\mathfrak {v}}^{\mathfrak {w}}$.
# Base change/Transformation matrix/Polynomial ring/Polynomials of degree 3/1/Exercise

Let $V$ be the vector space of all polynomials of
degree $\leq 3$, with the basis ${\mathfrak {u}}=X^{0},X^{1},X^{2},X^{3}$. Show that the polynomials
    $X^{3}+3X^{2}-X+4,\,-X^{3}+4X^{2}+5X+3,\,2X^{2}-X+1,\,3X^{3}+5X^{2}+7X-2$
form a basis of $V$, and determine the transformation matrices.
# Base change/Transformation matrix/2x2-matrices/1/Exercise

Let $V$ be the vector space of the $2\times 2$-matrices with the standard basis
    ${\mathfrak {u}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}},\,{\begin{pmatrix}0&1\\0&0\end{pmatrix}},\,{\begin{pmatrix}0&0\\1&0\end{pmatrix}},\,{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\,.$
Show that
    ${\mathfrak {v}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}},\,{\begin{pmatrix}1&0\\0&-1\end{pmatrix}},\,{\begin{pmatrix}0&1\\1&0\end{pmatrix}},\,{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}\,$
is also a basis of $V$, and determine the transformation matrices.