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.

1. Let ${}U_{j}$, ${}j\in J$, be a family of subspaces of ${}V$. Prove that also the intersection

   ${}U=\bigcap _{j\in J}U_{j}\,$

   is a subspace.

2. 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 ${}V$.
3. 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.

1. If the family is linearly independent, then for each subset ${}J\subseteq I$, also the family ${}v_{i}$ , ${}i\in J$ 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 ${}v$ is linearly independent if and only if ${}v\neq 0$.
6. 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

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