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

    Exercise for the break

### Exercise

Determine the dimension of the space of all $m\times n$-matrices.
    Exercises

### 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 $x,y,z,u,v$.

### 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.
An $n\times n$-matrix
    $M={\left(a_{ij}\right)}_{1\leq i,j\leq n}\,$
is called symmetric if $a_{ij}=a_{ji}$
$i,j$

### 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.

### 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.

### Exercise

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

### 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 $\dim _{K}{\left(U\right)}=\dim _{K}{\left(V\right)}$. Show that $U=V$ holds.

### 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 $0,1,2,3$.

### Exercise *

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 $V\times W$?

### 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 $U\cap V\neq 0$.

### Exercise

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

### Exercise

Show that the set of all real polynomials of degree $\leq 4$ that have a zero for $-2$ and for $3$, forms a finite-dimensional linear subspace in $\mathbb {R} [X]$. Determine its dimension.

### Exercise

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
    $v_{1},\ldots ,v_{n}{\text{ and }}{\mathrm {i} }v_{1},\ldots ,{\mathrm {i} }v_{n}$
forms a basis for $V$, considered as a real vector space.

### Exercise

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?

### Exercise

We consider the linear equations
    $9x-8y+7z-8u+4v=0\,,$
    $\,\,\,\,\,\,\,\,\,\,\,\,\,\,3y+7z-4u+6v=0\,,$
    $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-2z+5u+7v=0\,,$
over $\mathbb {R}$.
* Determine a basis ${\mathfrak {b}}_{1}$ of the solution space of this linear system.
* Extend the basis ${\mathfrak {b}}_{1}$ to a basis ${\mathfrak {b}}_{2}$ of the solution space of the linear system consisting of the first two equations.
* Extend 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.
* Extend the basis ${\mathfrak {b}}_{3}$ to a basis ${\mathfrak {b}}_{4}$ of the total space $\mathbb {R} ^{5}$.

### Exercise

We consider the last digit in the basic 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 dimension of the linear subspace in $\mathbb {R} ^{9}$ generated by these tuples?

### 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.
An $n\times n$-matrix ${\left(a_{ij}\right)}_{ij}$ over a 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.

### Exercise

Show that the set of all linear-magic squares of length $n$ over a field $K$ is a linear subspace in the space of all $n\times n$-matrices.
    Hand-in-exercises

### Exercise (2 marks)

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 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)

a) Determine the dimension of the solution space of the 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)

Show that the set of all real polynomials of degree $\leq 6$ that have a zero at $-1$, at $0$ and at $1$, is a finite-dimensional subspace of $\mathbb {R} [X]$. Determine the dimension of this vector space.

### Exercise (4 marks)

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

### Exercise (7 (3+2+1+1) marks)

We consider the linear equations
    $8x-3y+5z+7u+6v=0\,,$
    $9x+2y+\,\,z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-v=0\,,$
    $\,\,\,\,\,\,\,\,\,\,\,\,\,\,7y-\,\,z+4u\,\,\,\,\,\,\,\,\,\,\,\,\,\,=0\,,$
over $\mathbb {R}$.
* Determine a basis ${\mathfrak {b}}_{1}$ of the solution space of this linear system.
* Extend the basis ${\mathfrak {b}}_{1}$ to a basis ${\mathfrak {b}}_{2}$ of the solution space of the linear system consisting of the first two equations.
* Extend 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.
* Extend the basis ${\mathfrak {b}}_{3}$ to a basis ${\mathfrak {b}}_{4}$ of the total space $\mathbb {R} ^{5}$.