Linear algebra (Osnabrück 2024-2025)/Part I/Exercise sheet 7

Exercise for the break

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.

Exercises

Exercise

Find, for the vectors

${\begin{pmatrix}4\\5\end{pmatrix}},\,{\begin{pmatrix}1\\-4\end{pmatrix}},\,{\begin{pmatrix}8\\7\end{pmatrix}}$

in ${}\mathbb {Q} ^{2}$, a non-trivial representation of the zero vector.

Exercise

Find, for the vectors

${\begin{pmatrix}7\\-5\\3\end{pmatrix}},\,{\begin{pmatrix}-4\\1\\-6\end{pmatrix}},\,{\begin{pmatrix}2\\8\\0\end{pmatrix}},\,{\begin{pmatrix}5\\-5\\8\end{pmatrix}}$

in ${}\mathbb {Q} ^{3}$, a non-trivial representation of the zero vector.

Exercise

Decide whether the following vectors are linearly independent.

1. ${}(-1,1,-1)$, ${}(0,6,4)$, ${}(1,2,3)$, in the ${}\mathbb {R}$-vector space ${}\mathbb {R} ^{3}$.
2. ${}1+{\mathrm {i} }$, ${}1+2{\mathrm {i} }$ in the ${}\mathbb {R}$-vector space ${}\mathbb {C}$.
3. ${}1+{\mathrm {i} }$, ${}1+2{\mathrm {i} }$ in the ${}\mathbb {C}$-vector space ${}\mathbb {C}$.
4. ${}1$, ${}{\sqrt {3}}$ in the ${}\mathbb {Q}$-vector space ${}\mathbb {R}$.

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

Let ${}V$ be a ${}K$-vector space, and let ${}v_{1},\ldots ,v_{n}$ be a family of vectors in ${}V$. Show that the family is linearly independent if and only if there exists a linear subspace ${}U\subseteq V$ such that the family is a basis of ${}U$.

Exercise

Determine a basis for the linear subspace

${}U={\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x=z\right\}}\subset \mathbb {R} ^{3}\,.$

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

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

form 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

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

Exercise

In ${}\mathbb {R} ^{3}$, let the two linear subspaces

${}U={\left\{s{\begin{pmatrix}2\\1\\7\end{pmatrix}}+t{\begin{pmatrix}4\\-2\\9\end{pmatrix}}\mid s,t\in \mathbb {R} \right\}}\,$

and

${}V={\left\{p{\begin{pmatrix}3\\1\\0\end{pmatrix}}+q{\begin{pmatrix}5\\2\\-4\end{pmatrix}}\mid p,q\in \mathbb {R} \right\}}\,$

be given. Determine a basis for ${}U\cap V$.

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 ${}U\subseteq \mathbb {Q} ^{n}$ be a linear subspace. Show that ${}U$ has a basis consisting of vectors, such that all their entries are integer numbers.

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

Let ${}V$ be a ${}K$-vector space, let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}V$, and let

$\psi _{\mathfrak {v}}\colon K^{n}\longrightarrow V,{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n},$

be the corresponding bijective mapping in the sense of Remark 7.12 . Show that this mapping transforms the componentwise addition in ${}K^{n}$ into the vector addition in ${}V$, that is,

${}\psi _{\mathfrak {v}}{\left({\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}+{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}\right)}=\psi _{\mathfrak {v}}{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}+\psi _{\mathfrak {v}}{\begin{pmatrix}t_{1}\\\vdots \\t_{n}\end{pmatrix}}\,$

holds.

Exercise

Let ${}v_{1},\ldots ,v_{n}$ be a basis of ${}K^{n}$, and let

$\psi _{\mathfrak {v}}\colon K^{n}\longrightarrow K^{n},{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n},$

be the corresponding bijective mapping in the sense of Remark 7.12 . Show that this mapping is, in general, not compatible with componentwise multiplication in ${}K^{n}$.

Exercise

Let ${}V$ be a ${}K$-vector space, and let ${}v_{n}$, ${}n\in \mathbb {N} _{+}$, be a basis of ${}V$. Let ${}u_{n}$, ${}n\in \mathbb {N} _{+}$, be another family of vectors in ${}V$. Suppose that, for every ${}n\in \mathbb {N}$, the equality

${}\langle v_{1},\ldots ,v_{n}\rangle =\langle u_{1},\ldots ,u_{n}\rangle \,$

holds. Show that also ${}u_{n}$, ${}n\in \mathbb {N} _{+}$, is a basis of ${}V$.

Exercise

Let ${}\mathbb {R} [X]$ be the polynomial ring over ${}\mathbb {R}$. For ${}n\in \mathbb {N}$, set

${}F_{n}=1+2X+3X^{2}+\cdots +(n+1)X^{n}\,.$

Show that ${}F_{n}$, ${}n\in \mathbb {N}$, is a basis of ${}\mathbb {R} [X]$.

Exercise

Formulate and prove Theorem 7.11 for an arbitrary (not necessarily finite) family of vectors ${}v_{i}$, ${}i\in I$.

Exercise

We consider the real numbers ${}\mathbb {R}$ as a ${}\mathbb {Q}$-vector space. Show that the set of real numbers ${}\ln p$, where ${}p$ runs through the set of all prime numbers, is linearly independent. Tip: Use that every positive natural number has a unique representation as a product of prime numbers.

Exercise

What does the Theorem of Hamel mean in the following examples?

1. The real numbers ${}\mathbb {R}$ as a ${}\mathbb {Q}$-vector space.
2. The set of all real seqeunces

   ${}\mathbb {R} ^{\mathbb {N} }={\left\{{\left(x_{n}\right)}_{n\in \mathbb {N} }\mid x_{n}\in \mathbb {R} \right\}}\,.$

3. The set of all continuous functions from ${}\mathbb {R}$ to ${}\mathbb {R}$.

Exercise

Let ${}K$ be an ordered field, and let

${}V=K^{\mathbb {N} _{+}}\,$

be the vector space of all sequences in ${}K$ (with componentwise addition and scalar multiplication).

a) Show that (without using theorems about convergent sequences), the set of zero sequences, that is,

${}U={\left\{(x_{n})_{n\in \mathbb {N} _{+}}\mid (x_{n})_{n\in \mathbb {N} _{+}}{\text{ converges to 0}}\right\}}\,$

is a ${}K$-linear subspace of ${}V$.

b) Are the sequences

$(1/n)_{n\in \mathbb {N} _{+}}{\text{ and }}(1/n^{2})_{n\in \mathbb {N} _{+}}$

linearly independent in ${}V$?

Hand-in-exercises

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 (2 marks)

Show that, for the space of all ${}m\times n$-matrices ${}\operatorname {Mat} _{m\times n}(K)$, the matrices ${}E_{ij}$ that have in position ${}(i,j)$ the entry ${}1$, and elsewhere ${}0$, 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 an ${}\mathbb {R}$-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

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

Exercise (4 marks)

Let ${}\mathbb {R} [X]$ be the polynomial ring over ${}\mathbb {R}$. We set ${}P_{0}=1$, and, for ${}n\in \mathbb {N} _{+}$, we set

${}P_{n}=(X-1)(X-2)\cdots (X-n)\,.$

Show that ${}P_{n}$, ${}n\in \mathbb {N}$, is a basis of ${}\mathbb {R} [X]$.

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