# Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 7

    Warm-up-exercises

### Exercise

Write in $\mathbb {Q} ^{2}$ the vector
    $(2,-7)$
as a linear combination of the vectors
    $(5,-3){\text{ and }}(-11,4).$

### Exercise

Write in $\mathbb {C} ^{2}$ the vector
    $(1,0)$
as a linear combination of the vectors
    $(3+5{\mathrm {i} },-3+2{\mathrm {i} }){\text{ and }}(1-6{\mathrm {i} },4-{\mathrm {i} }).$

### Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_{i}$, $i\in I$, be a family of vectors in $V$, and let $w\in V$ be another vector. Assume that the family
    $w,v_{i},i\in I,$
is a system of generators of $V$, and that $w$ is a linear combination of the $v_{i}$, $i\in I$. Prove that also $v_{i}$, $i\in I$, is a system of generators of $V$.

### Exercise

Let $K$ be a field and let $V$ be a $K$-vector space. Prove the following facts.
* Let $U_{j}$, $j\in J$, be a family of subspaces of $V$. Prove that also the intersection is a subspace.
    $U=\bigcap _{j\in J}U_{j}\,$
* Let $v_{i}$, $i\in I$, be a family of elements of $V$ and consider the subset $W$ of $V$ which is given by all linear combinations of these elements. Show that $W$ is a subspace of $V$.
* The family $v_{i}$, $i\in I$, is a system of generators of $V$ if and only if
    $\langle v_{i},\,i\in I\rangle =V\,.$

### Exercise

Show that the three vectors
    ${\begin{pmatrix}0\\1\\2\\1\end{pmatrix}},\,{\begin{pmatrix}4\\3\\0\\2\end{pmatrix}},\,{\begin{pmatrix}1\\7\\0\\-1\end{pmatrix}}$
in $\mathbb {R} ^{4}$ are linearly independent.

### Exercise

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

### Exercise

Let $K$ be a field, let $V$ be a $K$-vector space and let $v_{i}$, $i\in I$, be a family of vectors in $V$. Prove the following facts.
* If the family is linearly independent, then for each subset $J\subseteq I$, also the family $v_{i}$ , $i\in J$ is linearly independent.
* The empty family is linearly independent.
* If the family contains the null vector, then it is not linearly independent.
* If a vector appears several times in the family, then the family is not linearly independent.
* A vector $v$ is linearly independent if and only if $v\neq 0$.
* Two vectors $v$ and $u$ are linearly independent if and only if $u$ is not a scalar multiple of $v$ and vice versa.

### Exercise

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

### Exercise

Determine a basis for the solution space of the linear equation
    $3x+4y-2z+5w=0\,.$

### Exercise

Determine a basis for the solution space of the linear system of equations
    $-2x+3y-z+4w=0{\text{ and }}3z-2w=0.$

### Exercise

Prove that in $\mathbb {R} ^{3}$, the three vectors
    ${\begin{pmatrix}2\\1\\5\end{pmatrix}}\,,{\begin{pmatrix}1\\3\\7\end{pmatrix}}\,,{\begin{pmatrix}4\\1\\2\end{pmatrix}}$
are a basis.

### Exercise

Establish if in $\mathbb {C} ^{2}$ the two vectors
    ${\begin{pmatrix}2+7{\mathrm {i} }\\3-{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}15+26{\mathrm {i} }\\13-7{\mathrm {i} }\end{pmatrix}}$
form a basis.

### Exercise

Let $K$ be a field. Find a linear system of equations in three variables, whose solution space is exactly
    ${\left\{\lambda {\begin{pmatrix}3\\2\\-5\end{pmatrix}}\mid \lambda \in K\right\}}.$
    Hand-in-exercises

### Exercise (3 marks)

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

### Exercise (2 marks)

Establish if in $\mathbb {R} ^{3}$ the three vectors
    ${\begin{pmatrix}2\\3\\-5\end{pmatrix}}\,,{\begin{pmatrix}9\\2\\6\end{pmatrix}}\,,{\begin{pmatrix}-1\\4\\-1\end{pmatrix}}$
form a basis.

### Exercise (2 marks)

Establish if in $\mathbb {C} ^{2}$ the two vectors
    ${\begin{pmatrix}2-7{\mathrm {i} }\\-3+2{\mathrm {i} }\end{pmatrix}}{\text{ and }}{\begin{pmatrix}5+6{\mathrm {i} }\\3-17{\mathrm {i} }\end{pmatrix}}$
form a basis.

### Exercise (4 marks)

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

### Exercise (3 marks)

Let $K$ be a field, and let
    ${\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\in K^{n}\,$
be a nonzero vector. Find a linear system of equations in $n$ variables with $n-1$ equations, whose solution space is exactly
    ${\left\{\lambda {\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}\mid \lambda \in K\right\}}.$