Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Exercise sheet 23

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 ${ℂ}^2$ 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. Show that the following statements hold.

1. For a family $v_i$, $i\in I$, of elements in $V$, linear span is a linear subspace of $V$.
2. The family $v_i$, $i\in I$, is a spanning system of $V$ if and only if

   $⟨v_i,i\in I⟩=V.$

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 $w_j$, $j\in J$, another family of vectors in $V$. Then, for the spanned linear subspaces, the inclusion $⟨v_i,i\in I⟩\subseteq ⟨w_j,j\in J⟩$ holds, if and only if $v_i\in ⟨w_j,j\in J⟩$ holds for all $i\in I$.

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

We consider in ${\mathbb{Q}}^3$ the linear subspaces

$U=⟨\left(\begin{array}{c}2\\ 1\\ 4\end{array}\right),\left(\begin{array}{c}3\\ -2\\ 7\end{array}\right)⟩$

and

$W=⟨\left(\begin{array}{c}5\\ -1\\ 11\end{array}\right),\left(\begin{array}{c}1\\ -3\\ 3\end{array}\right)⟩.$

Show that $U=W$.

Exercise

Show that the three vectors

$\left(\begin{array}{c}0\\ 1\\ 2\\ 1\end{array}\right),\left(\begin{array}{c}4\\ 3\\ 0\\ 2\end{array}\right),\left(\begin{array}{c}1\\ 7\\ 0\\ -1\end{array}\right)$

in ${ℝ}^4$ are linearly independent.

Exercise

Find, for the vectors

$\left(\begin{array}{c}7\\ -5\\ 3\end{array}\right),\left(\begin{array}{c}-4\\ 1\\ -6\end{array}\right),\left(\begin{array}{c}2\\ 8\\ 0\end{array}\right),\left(\begin{array}{c}5\\ -5\\ 8\end{array}\right)$

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

Exercise

Give an example of three vectors in ${ℝ}^3$ 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 $v\ne 0$.
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 $\ne 0$ 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 ${ℝ}^3$, the three vectors

$\left(\begin{array}{c}2\\ 1\\ 5\end{array}\right),\left(\begin{array}{c}1\\ 3\\ 7\end{array}\right),\left(\begin{array}{c}4\\ 1\\ 2\end{array}\right)$

form a basis.

Exercise

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

$\left(\begin{array}{c}2+7\mathrm{i}\\ 3-\mathrm{i}\end{array}\right)\text{ and }\left(\begin{array}{c}15+26\mathrm{i}\\ 13-7\mathrm{i}\end{array}\right)$

form a basis.

Exercise

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

$\{\lambda \left(\begin{array}{c}3\\ 2\\ -5\end{array}\right)\mid \lambda \in K\}.$

Exercise

Let $K$ be a field, and let

$\left(\begin{array}{c}{a}_1\\ ⋮\\ a_n\end{array}\right)\in K^n$

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

$\{\lambda \left(\begin{array}{c}{a}_1\\ ⋮\\ a_n\end{array}\right)\mid \lambda \in K\}.$

Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space of dimension $n={\dim }_K\left(V\right)$. Suppose that $n$ vectors $v_1,\dots ,v_n$ in $V$ are given. Prove that the following facts are equivalent.

1. $v_1,\dots ,v_n$ form a basis for $V$.
2. $v_1,\dots ,v_n$ form a system of generators for $V$.
3. $v_1,\dots ,v_n$ are linearly independent.

Exercise

Let $K$ be a field, and let $K[X]$ denote the polynomial ring over $K$. Let $d\in \mathbb{N}$. Show that the set of all polynomials of degree $\le d$ is a finite-dimensional linear subspace of $K[X]$. What is its dimension?

Exercise

Show that the set of all real polynomials of degree $\le 4$ that have a zero for $-2$ and for $3$, forms a finite-dimensional linear subspace in $ℝ[X]$. Determine its dimension.

Exercise *

Let $K$ be a field, and let $V$ and $W$ be two finite-dimensional $K$-vector spaces with

${\dim }_K\left(V\right)=n$

and

${\dim }_K\left(W\right)=m.$

What is the dimension of the Cartesian product $V\times W$?

Exercise

Let $V$ be a finite-dimensional vector space over the complex numbers, and let $v_1,\dots ,v_n$ be a basis of $V$. Prove that the family of vectors

$v_1,\dots ,v_n\text{ and }\mathrm{i}{v}_1,\dots ,\mathrm{i}{v}_n$

forms a basis for $V$, considered as a real vector space.

Exercise

Let $K$ be a finite field with $q$ elements, and let $V$ be an $n$-dimensional vector space. Let $v_1,v_2,v_3,\dots$ be an enumeration (without repetitions) of the elements from $V$. After how many elements can we be sure that these form a generating system of $V$.

Hand-in-exercises

Exercise (3 marks)

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

$(2,5,-3)$

as a linear combination of the vectors

$(1,2,3),(0,1,1),\text{ and }(-1,2,4).$

Prove that it cannot be expressed as a linear combination of two of the three vectors.

Exercise (4 marks)

We consider in ${\mathbb{Q}}^4$ the linear subspaces

$U=⟨\left(\begin{array}{c}3\\ 1\\ -5\\ 2\end{array}\right),\left(\begin{array}{c}2\\ -2\\ 4\\ -3\end{array}\right),\left(\begin{array}{c}1\\ 0\\ 3\\ 2\end{array}\right)⟩$

and

$W=⟨\left(\begin{array}{c}6\\ -1\\ 2\\ 1\end{array}\right),\left(\begin{array}{c}0\\ -2\\ -2\\ -7\end{array}\right),\left(\begin{array}{c}9\\ 2\\ -1\\ 10\end{array}\right)⟩.$

Show that $U=W$.

Exercise (2 marks)

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

$\left(\begin{array}{c}2\\ 3\\ -5\end{array}\right),\left(\begin{array}{c}9\\ 2\\ 6\end{array}\right),\left(\begin{array}{c}-1\\ 4\\ -1\end{array}\right)$

form a basis.

Exercise (2 marks)

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

$\left(\begin{array}{c}2-7\mathrm{i}\\ -3+2\mathrm{i}\end{array}\right)\text{ and }\left(\begin{array}{c}5+6\mathrm{i}\\ 3-17\mathrm{i}\end{array}\right)$

form a basis.

Exercise (4 marks)

Let ${\mathbb{Q}}^n$ be the $n$-dimensional standard vector space over $\mathbb{Q}$, and let $v_1,\dots ,v_n\in {\mathbb{Q}}^n$ 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 ${ℝ}^n$, is an $ℝ$-basis of ${ℝ}^n$.

Exercise (4 marks)

Show that the set of all real polynomials of degree $\le 6$ that have a zero at $-1$, at $0$ and at $1$, is a finite-dimensional subspace of $ℝ[X]$. Determine the dimension of this vector space.

Exercise (2 marks)

Let $K$ be a field, and let $V$ be a $K$-vector space. Let $v_1,\dots ,v_m$ be a family of vectors in $V$, and let

$U=⟨v_i,i=1,\dots ,m⟩$

be the linear subspace they span. Prove that the family is linearly independent if and only if the dimension of $U$ is exactly $m$.

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