# Kurs:Mathematik für Anwender (Osnabrück 2011-2012)/Teil I/Arbeitsblatt 7/en

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Warm-up-exercises

### Exercise

Write in ${\displaystyle {}\mathbb {Q} ^{2}}$ the vector

${\displaystyle (2,-7)}$
as a linear combination of the vectors
${\displaystyle (5,-3){\text{ and }}(-11,4).}$

### Exercise

Write in ${\displaystyle {}\mathbb {C} ^{2}}$ the vector

${\displaystyle (1,0)}$
as a

linear combination of the vectors

${\displaystyle (3+5i,-3+2i){\text{ and }}(1-6i,4-i).}$

### Exercise

Let ${\displaystyle {}K}$ be a field and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. Let ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, be a family of vectors in ${\displaystyle {}V}$ and let ${\displaystyle {}w\in V}$ be another vector. Assume that the family

${\displaystyle w,v_{i},i\in I,}$
is a system of generators

of ${\displaystyle {}V}$ and that ${\displaystyle {}w}$ is a linear combination of the ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$. Prove that also ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, is a system of generators of ${\displaystyle {}V}$.

### Exercise

Show that the three vectors

${\displaystyle {\begin{pmatrix}0\\1\\2\\1\end{pmatrix}},\,{\begin{pmatrix}4\\3\\0\\2\end{pmatrix}},\,{\begin{pmatrix}1\\7\\0\\-1\end{pmatrix}}}$

in ${\displaystyle {}\mathbb {R} ^{4}}$ are linearly independent.

### Exercise

Give an example of three vectors in ${\displaystyle {}\mathbb {R} ^{3}}$ such that each couple of them is linearly independent, but all three vectors are linearly dependent.

### Exercise

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space and let ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, be a family of vectors in ${\displaystyle {}V}$. Prove the following facts.

1. If the family is linearly independent then for each subset ${\displaystyle {}J\subseteq I}$ also the family ${\displaystyle {}v_{i}}$ , ${\displaystyle {}i\in J}$ is linearly independent.
2. The empty family is linearly independent.
3. If the family contains the null vector then it is not linearly independent.
4. If a vector appears several times in the family, then the family is not linearly independent.
5. A vector ${\displaystyle {}v}$ is linearly independent if and only if ${\displaystyle {}v\neq 0}$.
6. Two vectors ${\displaystyle {}v}$ und ${\displaystyle {}u}$ are linearly independent if and only if ${\displaystyle {}u}$ is not a scalar multiple of ${\displaystyle {}v}$ and vice versa.

### Exercise

Let ${\displaystyle {}K}$ be a field, let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space and let ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, be a family of vectors in ${\displaystyle {}V}$. Let ${\displaystyle {}\lambda _{i}}$, ${\displaystyle {}i\in I}$ be a family of elements ${\displaystyle {}\neq 0}$ in ${\displaystyle {}K}$. Prove that the family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, is linearly independent (a system of generators of ${\displaystyle {}V}$, a basis of ${\displaystyle {}V}$) if and only if the same holds for the family ${\displaystyle {}\lambda _{i}v_{i}}$, ${\displaystyle {}i\in I}$.

### Exercise

Determine a basis for the solution space of the linear equation

${\displaystyle 3x+4y-2z+5w=0.}$

### Exercise

Determine a basis for the solution space of the linear system of equations

${\displaystyle -2x+3y-z+4w=0{\text{ and }}3z-2w=0.}$

### Exercise

Prove that in ${\displaystyle {}\mathbb {R} ^{3}}$ the three vectors

${\displaystyle {\begin{pmatrix}2\\1\\5\end{pmatrix}}\,,{\begin{pmatrix}1\\3\\7\end{pmatrix}}\,,{\begin{pmatrix}4\\1\\2\end{pmatrix}}}$

are a basis.

### Exercise

Establish if in ${\displaystyle {}\mathbb {C} ^{2}}$ the two vectors

${\displaystyle {\begin{pmatrix}2+7i\\3-i\end{pmatrix}}{\text{ and }}{\begin{pmatrix}15+26i\\13-7i\end{pmatrix}}}$

form a basis.

### Exercise

Let ${\displaystyle {}K}$ be a field. Find a linear system of equations in three variables, whose solution space is exactly

${\displaystyle {\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.}$

Hand-in-exercises

### Exercise (3 points)

Write in ${\displaystyle {}\mathbb {Q} ^{3}}$ the vector

${\displaystyle (2,5,-3)}$
as a linear combination of the vectors
${\displaystyle (1,2,3),(0,1,1){\text{ und }}(-1,2,4).}$
Prove that it cannot be expressed as a linear combination of two of the three vectors.

### Exercise (2 points)

Establish if in ${\displaystyle {}\mathbb {R} ^{3}}$ the three vectors

${\displaystyle {\begin{pmatrix}2\\3\\-5\end{pmatrix}}\,,{\begin{pmatrix}9\\2\\6\end{pmatrix}}\,,{\begin{pmatrix}-1\\4\\-1\end{pmatrix}}}$

form a basis.

### Exercise (2 points)

Establish if in ${\displaystyle {}\mathbb {C} ^{2}}$ the two vectors

${\displaystyle {\begin{pmatrix}2-7i\\-3+2i\end{pmatrix}}{\text{ and }}{\begin{pmatrix}5+6i\\3-17i\end{pmatrix}}}$

form a basis.

### Exercise (4 points)

Let ${\displaystyle {}\mathbb {Q} ^{n}}$ be the ${\displaystyle {}n}$-dimensional standard vector space over ${\displaystyle {}\mathbb {Q} }$ and let ${\displaystyle {}v_{1},\ldots ,v_{n}\in \mathbb {Q} ^{n}}$ be a family of vectors. Prove that this family is a ${\displaystyle {}\mathbb {Q} }$-basis of ${\displaystyle {}\mathbb {Q} ^{n}}$ if and only if the same family, considered as a family in ${\displaystyle {}\mathbb {R} ^{n}}$, is a ${\displaystyle {}\mathbb {R} }$-basis of ${\displaystyle {}\mathbb {R} ^{n}}$.

### Exercise (3 points)

Let ${\displaystyle {}K}$ be a field and let

${\displaystyle {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}}$

be a nonzero vector. Find a linear system of equations in ${\displaystyle {}n}$ variables with ${\displaystyle {}n-1}$ equations, whose solution space is exactly

${\displaystyle {\left\{\lambda {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\mid \lambda \in K\right\}}.}$