# Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 5/refcontrol

    Exercise for the break

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Determine an equation for the line in $\mathbb {R} ^{2}$ which runs through the points $(-4,3)$ and $(5,-6)$.
    Exercises

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Solve the following system of inhomogeneous linear equations.
    ${\begin{matrix}3x&\,\,\,\,\,\,\,\,&+z&+4w&=&4\\2x&+2y&\,\,\,\,\,\,\,\,&+w&=&0\\4x&+6y&\,\,\,\,\,\,\,\,&+w&=&2\\x&+3y&+5z&\,\,\,\,\,\,\,\,&=&3\,.\end{matrix}}$

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Determine a linear equation for the plane in $\mathbb {R} ^{3}$, where the three points
    $(1,0,0),\,(0,1,2){\text{ and }}(2,3,4)$
lie.

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Bring the system of linear equations
    $3x-4+5y=8z+7x\,,$
    $2-4x+z=2y+3x+6\,,$
    $4z-3x+2x+3=5x-11y+2z-8\,$
into a standard form, and solve it.

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Solve, over the complex numbers,MDLD/complex numbers the linear systemMDLD/linear system of equations
    ${\begin{matrix}{\mathrm {i} }x&+y&+(2-{\mathrm {i} })z&=&2\\&7y&+2{\mathrm {i} }z&=&-1+3{\mathrm {i} }\\&&(2-5{\mathrm {i} })z&=&1\,.\end{matrix}}$

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Let $K$ be the field with two elements. Solve in $K$ the inhomogeneous linear systemMDLD/inhomogeneous linear system
    ${\begin{matrix}x&+y&&=&1\\&y&+z&=&0\\x&+y&+z&=&0\,.\end{matrix}}$

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Given a
complex numberMDLD/complex number
    $z=a+b{\mathrm {i} }\neq 0\,,$
find its inverse complex number with the help of a real system of linear equations, with two equations in two variables.
The field $\mathbb {Q} [{\sqrt {3}}]$ consists of all real numbers of the form $a+b{\sqrt {3}}$ with $a,b\in \mathbb {Q}$. The inverse element of $a+b{\sqrt {3}}\neq 0$ is ${\frac {a}{a^{2}+3b^{2}}}-{\frac {b}{a^{2}+3b^{2}}}{\sqrt {3}}$.

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

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Solve the linear system
    ${\begin{matrix}4y&+7z&+3&=&4\\11y&+9z&+13&=&9\\6y&+8z&+5&=&2\,\end{matrix}}$
with the substitution method.MDLD/substitution method

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Solve the linear system
    ${\begin{matrix}5y&+6z&+2&=&6\\4y&+8z&+9&=&5\\11y&+5z&+7&=&8\,\end{matrix}}$
with the equating method.MDLD/equating method

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* We consider the linear system over $\mathbb {Q}$, consisting in the two equations and Determine a linear system, which is equivalent to the given system and has the property that all coefficients are integers.
    ${\frac {3}{7}}x+{\frac {4}{9}}y-{\frac {3}{15}}z={\frac {1}{35}}\,$
    $-{\frac {5}{3}}x+{\frac {1}{4}}y-{\frac {6}{7}}z={\frac {11}{10}}\,.$
* Show that for every system of linear equationsMDLD/system of linear equations over $\mathbb {Q}$, there exists an equivalentMDLD/equivalent (linear system) linear system with the property that all coefficients are integers.

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Show that, for every linear systemMDLD/linear system over $\mathbb {Q}$, there exists an equivalentMDLD/equivalent (linear system) system with the property that the modulus of all coefficients is smaller than $1$.

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Show with an example that the linear system given by three equations I, II, III is not equivalent to the linear system given by the three equations I-II, I-III, II-III.

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Out of the resources $R_{1},R_{2}$ and $R_{3}$, several commodities $P_{1},P_{2},P_{3},P_{4}$ are produced. The following table shows, how much of the resources is needed to produce the commodities (always in suitable units).
a) Establish a matrix, which computes, applied to a four-tuple of commodities, the required resources.
b) The following table shows how much of each commodity shall be produced in a month.
What resources are necessary?
c) The following table shows how much of each resource is delivered on a certain day.
Which tuples of commodities can be produced from this without waste?

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Let
    $D={\begin{pmatrix}d_{11}&0&\cdots &\cdots &0\\0&d_{22}&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&d_{n-1\,n-1}&0\\0&\cdots &\cdots &0&d_{nn}\end{pmatrix}}\,$
be a diagonal matrix,MDLD/diagonal matrix and $c={\begin{pmatrix}c_{1}\\\vdots \\c_{n}\end{pmatrix}}$ an $n$-tuple over a fieldMDLD/field $K$, and let $x={\begin{pmatrix}x_{1}\\\vdots \\x_{n}\end{pmatrix}}$ be a tuple of variables. What is specific about the system of linear equations
    $Dx=c\,,$
and how can you solve it?

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Solve the linear systemsMDLD/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.

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A system of linear inequalities is given by
    $x\geq 0\,,$
    $y\geq 0\,,$
    $x+y\leq 1\,$
Sketch the solution set of this system of inequalities.

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Let
    $a_{1}x+b_{1}y\geq c_{1}\,,$
    $a_{2}x+b_{2}y\geq c_{2}\,,$
    $a_{3}x+b3y\geq c_{3}\,,$
be a system of linear inequalities, whose solution set is a triangle. How does the solution set look like, when we replace one inequality $\geq$ by $\leq$?
===Exercise Exercise 5.19
Prove the superposition principle for systems of linear equations.

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Determine, in dependence of the parameter $a\in \mathbb {R}$, the solution space $L_{a}\subseteq \mathbb {R} ^{3}$ of the system of linear equations
    $5x+ay+(1-a)z=0\,,$
    $2ax+a^{2}y+3z=0\,.$
    Hand-in-exercises

### Exercise (4 marks) Create referencenumber

Solve the following system of inhomogeneous linear equations.
    ${\begin{matrix}x&+2y&+3z&+4w&=&1\\2x&+3y&+4z&+5w&=&7\\x&\,\,\,\,\,\,\,\,&+z&\,\,\,\,\,\,\,\,&=&9\\x&+5y&+5z&+w&=&0\,.\end{matrix}}$

### Exercise (3 marks) Create referencenumber

Consider in $\mathbb {R} ^{3}$ the two planes
    $E={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 3x+4y+5z=2\right\}}{\text{ and }}F={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid 2x-y+3z=-1\right\}}.$
Determine the intersecting line $E\cap F$.

### Exercise (3 marks) Create referencenumber

Determine a linear equation for the plane in $\mathbb {R} ^{3}$, where the three points
    $(1,0,2),\,\,(4,-3,2)\,{\text{ and }}\,(2,1,-1)$
lie.

### Exercise (4 marks) Create referencenumber

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?

### Exercise (3 marks) Create referencenumber

Solve the linear system
    ${\begin{matrix}5y&\,\,\,\,-z&+7&=&6\\3y&+6z&+3&=&-2\\8y&+8z&+7&=&3\,\end{matrix}}$
with the solution method.MDLD/solution method

### Exercise (4 (2+2) marks) Create referencenumber

A system of linear inequalities is given by
    $x\geq 0\,,$
    $y+x\geq 0\,,$
    $-1-y\leq -x\,,$
    $5y-2x\geq 3\,.$
a) Sketch the solution set of this system.
b) Determine the corners for this solution set.