Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 8/refcontrol

Exercise for the break

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Determine the dimensionMDLD/dimension (vs) of the space of all ${}m\times n$-matrices.MDLD/matrices

Exercises

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Determine the dimensionMDLD/dimension (vs) of the solution space of the linear systemMDLD/linear system

${}4x-3y+7z+5u-v=0\,$

and

${}y+6z-10u+3v=0\,$

in the variables ${}x,y,z,u,v$.

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Let ${}K$ be a field,MDLD/field and ${}n\in \mathbb {N}$. Show that the set of all diagonal matricesMDLD/diagonal matrices is a linear subspaceMDLD/linear subspace in the space of all ${}n\times n$-matricesMDLD/matrices over ${}K$, and determine its dimension.MDLD/dimension (vs)

An ${}n\times n$-matrixMDLD/matrix

${}M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,$

is called symmetric, if ${}a_{ij}=a_{ji}$

holds for all ${}i,j$.

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Show that the set of all symmetricMDLD/symmetric (matrix) ${}n\times n$-matricesMDLD/matrices is a linear subspaceMDLD/linear subspace in the space of all ${}n\times n$-matrices, and determine its dimension.MDLD/dimension (vs)

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Let ${}K$ be a field,MDLD/field and ${}n\in \mathbb {N}$. Show that the set of all upper triangular matricesMDLD/upper triangular matrices is a linear subspaceMDLD/linear subspace in the space of all ${}n\times n$-matricesMDLD/matrices over ${}K$, and determine its dimension.MDLD/dimension (vs)

===Exercise Exercise 8.6

change===

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space of dimension ${}n=\dim _{K}{\left(V\right)}$. Suppose that ${}n$ vectors ${}v_{1},\ldots ,v_{n}$ in ${}V$ are given. Prove that the following facts are equivalent.

1. ${}v_{1},\ldots ,v_{n}$ form a basis for ${}V$.
2. ${}v_{1},\ldots ,v_{n}$ form a system of generators for ${}V$.
3. ${}v_{1},\ldots ,v_{n}$ are linearly independent.

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Let ${}K$ be a field,MDLD/field and let ${}V$ be a ${}K$-vector spaceMDLD/vector space of finite dimension.MDLD/dimension (fgvs) Let ${}U\subseteq V$ denote a linear subspaceMDLD/linear subspace with ${}\dim _{K}{\left(U\right)}=\dim _{K}{\left(V\right)}$. Show that ${}U=V$ holds.

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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 ${}0,1,2,3$.

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Let ${}K$ be a field, and let ${}V$ and ${}W$ be two finite-dimensional ${}K$vector spaces with

${}\dim _{K}{\left(V\right)}=n\,$

and

${}\dim _{K}{\left(W\right)}=m\,.$

What is the dimension of the Cartesian product ${}V\times W$?

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

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Let ${}K$ be a field, and let ${}K[X]$ denote the polynomial ringMDLD/polynomial ring (field 1) over ${}K$. Let ${}d\in \mathbb {N}$. Show that the set of all polynomials of degree ${}\leq d$ is a finite-dimensionalMDLD/finite-dimensional (vs) linear subspaceMDLD/linear subspace of ${}K[X]$. What is its dimension?MDLD/dimension (vs)

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Show that the set of all real polynomialsMDLD/polynomials (field 1) of degreeMDLD/degree (polynomial) ${}\leq 4$, which have a zero for ${}-2$ and for ${}3$, form a finite-dimensionalMDLD/finite-dimensional linear subspaceMDLD/linear subspace in ${}\mathbb {R} [X]$. Determine its dimension.MDLD/dimension (vs)

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Let ${}V$ be a finite-dimensional vector space over the complex numbers, and let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$. Prove that the family of vectors

$v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}$

form a basis for ${}V$, considered as a real vector space.

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Consider the standard basis ${}e_{1},e_{2},e_{3},e_{4}$ in ${}\mathbb {R} ^{4}$ and the three vectors

${\begin{pmatrix}1\\3\\0\\-4\end{pmatrix}},\,{\begin{pmatrix}2\\1\\5\\7\end{pmatrix}}{\text{ and }}{\begin{pmatrix}-4\\9\\-5\\1\end{pmatrix}}.$

Prove that these vectors are linearly independent, and extend them to a basis by adding an appropriate standard vector as shown in the base exchange theorem. Can one take any standard vector?

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We consider the linear equationsMDLD/linear equations

${}9x-8y+7z-8u+4v=0\,,$

${}\,\,\,\,\,\,\,\,\,\,\,\,\,\,3y+7z-4u+6v=0\,,$

${}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2z+5u+7v=0\,$

over ${}\mathbb {R}$.

1. Determine a basisMDLD/basis (vs) ${}{\mathfrak {b}}_{1}$ of the solution space of this linear system.
2. Complete the basis ${}{\mathfrak {b}}_{1}$ to a basis ${}{\mathfrak {b}}_{2}$ of the solution space of the linear system consisting in the first two equations.
3. Complete the basis ${}{\mathfrak {b}}_{2}$ to a basis ${}{\mathfrak {b}}_{3}$ of the solution space of the linear system consisting of the first equation alone.
4. Complete the basis ${}{\mathfrak {b}}_{3}$ to a basis ${}{\mathfrak {b}}_{4}$ of the total space ${}\mathbb {R} ^{5}$.

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We consider the last digit in the basis multiplication table as a family of ${}9$-tuples of length ${}9$, that is, the row vectors in the matrix

${\begin{pmatrix}1&2&3&4&5&6&7&8&9\\2&4&6&8&0&2&4&6&8\\3&6&9&2&5&8&1&4&7\\4&8&2&6&0&4&8&2&6\\5&0&5&0&5&0&5&0&5\\6&2&8&4&0&6&2&8&4\\7&4&1&8&5&2&9&6&3\\8&6&4&2&0&8&6&4&2\\9&8&7&6&5&4&3&2&1\end{pmatrix}}.$

What is the dimensionMDLD/dimension (vs) of the linear subspaceMDLD/linear subspace in ${}\mathbb {R} ^{9}$ generated by these tuples?

===Exercise Exercise 8.17

change===

Let ${}K$ be a field,MDLD/field and let ${}V$ be a ${}K$-vector space.MDLD/vector space Show that ${}V$ can not have a finite basisMDLD/basis (vs) and an infinite basis.

The magic square in Dürer's picture Melencolia I.

An ${}n\times n$-matrixMDLD/matrix ${}{\left(a_{ij}\right)}_{ij}$ over a fieldMDLD/field ${}K$ is called a magic square (or linear-magic square over ${}K$), when every column sum and every row sum in the matrix equals a certain number

${}c\in K$.

In this sense, the matrix

${\begin{pmatrix}c&0&\cdots &0\\0&c&\ddots &\vdots \\\vdots &\ddots &\ddots &0\\0&\cdots &0&c\end{pmatrix}}$

is, for every ${}c\in K$, a magic square.

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Show that the set of all linear-magic squaresMDLD/linear-magic squares of length ${}n$ over a fieldMDLD/field ${}K$ is a linear subspaceMDLD/linear subspace in the space of all ${}n\times n$-matrices.MDLD/matrices

Hand-in-exercises

Exercise (2 marks) Create referencenumber

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{1},\ldots ,v_{m}$ be a family of vectors in ${}V$, and let

${}U=\langle v_{i},\,i=1,\ldots ,m\rangle \,$

be the linear subspaceMDLD/linear subspace they span. Prove that the family is linearly independent if and only if the dimension of ${}U$ is exactly ${}m$.

Exercise (4 (3+1) marks) Create referencenumber

a) Determine the dimensionMDLD/dimension (vs) of the solution space of the linear systemMDLD/linear system

${}2x+5y+7z+4u-3v+2w=0\,$

${}4x+9y+6z+5u-v+w=0\,$

${}7x+8y-3z+u+3v+3w=0\,$

${}-x+6y+16z+8u-7v=0\,$

in the variables ${}x,y,z,u,v,w$.

b) What is the dimension of the solution space, if we consider the system in the variables ${}x,y,z,u,v,w,r,s$?

Exercise (4 marks) Create referencenumber

Show that the set of all real polynomialsMDLD/polynomials (field 1) of degreeMDLD/degree (polynomial) ${}\leq 6$, which have a zero at ${}-1$, at ${}0$ and at ${}1$, is a finite-dimensionalMDLD/finite-dimensional subspaceMDLD/subspace (linear) of ${}\mathbb {R} [X]$. Determine the dimensionMDLD/dimension (vs) of this vector space.

Exercise (4 marks) Create referencenumber

Let ${}K$ be a field,MDLD/field and ${}n\in \mathbb {N} _{+}$. Determine the dimensionMDLD/dimension (vs) of the space of all linear-magic squaresMDLD/linear-magic squares of length ${}n$ over ${}K$.

Exercise (7 (3+2+1+1) marks) Create referencenumber

We consider the linear equationsMDLD/linear equations

${}8x-3y+5z+7u+6v=0\,,$

${}9x+2y+\,\,z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-v=0\,,$

${}\,\,\,\,\,\,\,\,\,\,\,\,\,\,7y-\,\,z+4u\,\,\,\,\,\,\,\,\,\,\,\,\,\,=0\,$

over ${}\mathbb {R}$.

1. Determine a basisMDLD/basis (vs) ${}{\mathfrak {b}}_{1}$ of the solution space of this linear system.
2. Complete the basis ${}{\mathfrak {b}}_{1}$ to a basis ${}{\mathfrak {b}}_{2}$ of the solution space of the linear system consisting in the first two equations.
3. Complete the basis ${}{\mathfrak {b}}_{2}$ to a basis ${}{\mathfrak {b}}_{3}$ of the solution space of the linear system consisting of the first equation alone.
4. Complete the basis ${}{\mathfrak {b}}_{3}$ to a basis ${}{\mathfrak {b}}_{4}$ of the total space ${}\mathbb {R} ^{5}$.

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