Vector spaces

$\alpha x=0$ if and only if $\alpha =0$ or $x=0$;

$\alpha x+\beta y=\beta x+\alpha y$ if and only if $\alpha =\beta$ or $x=y$.

does linear independence of the system of vectors $\left\{ a_{1}, a_{2}\right\}$ imply linear independence of the system $\left\{ \lambda a_{1}+a_{2}, a_{1}+\lambda a_{2}\right\}$;

does linear independence of the system $\left\{ a_{1}, \ldots , a_{n}\right\}$ imply linear independence of the system

$\sin x, \cos x$;

$1, \sin x, \cos x$;

$\sin x, \sin 2 x, \ldots , \sin n x ;$

$1, \cos x, \cos 2 x, \ldots , \cos n x$;

$1, \cos x, \sin x, \cos 2 x, \sin 2 x, \ldots , \cos n x, \sin n x$;

$1, \sin x, \sin^{2} x, \ldots , \sin^{n} x$;

$1, \cos x, \cos^{2} x, \ldots , \cos^{n} x$.

$e^{\alpha_{1} x}, e^{\alpha_{2} x}, \ldots , e^{\alpha_{n} x}$;

$x^{\alpha_{1}}, x^{\alpha_{2}}, \ldots , x^{\alpha_{n}}$;

$\left(1-\alpha_{1} x\right)^{-1}, \ldots ,\left(1-\alpha_{n} x\right)^{-1}$; where $\alpha_{1}, \ldots , \alpha_{n}$ are pairwise distinct real numbers.

In the vector space $V$ over the field $\mathbb {C}$, define a new multiplication of vectors by complex numbers by the rule $\alpha \circ x=\bar{\alpha } x$. Prove that with respect to the operations + and $\circ$, $V$ is again a vector space. Find its dimension.

Let $\mathbb {C}^{n}$ be the abelian group of all rows $(a_{1}, \ldots , a_{n})$ of length $n$, $a_{i} \in \mathbb {C}$. For $b \in \mathbb {C}$, set $b \circ \left(a_{1}, \ldots , a_{n}\right)=\left(b \bar{a}_{1}, \ldots , b \bar{a}_{n}\right)$. Is $\mathbb {C}^{n}$ a vector space with respect to the operations + and $\circ$?

the group $\mathbb {Z}$ is not isomorphic to the additive group of any vector space;

the group of residues $\mathbf{Z}_{n}$ is isomorphic to the additive group of a vector space over some field if and only if $n$ is prime;

a commutative group $A$ can be turned into a vector space over the field $\mathbf{Z}_{p}$ if and only if $p x=0$ for every $x \in A$;

a commutative group $A$ can be turned into a vector space over the field $\mathbb {Q}$ if and only if it has no elements of finite order (other than zero) and for every natural number $n$ and every $a \in A$ the equation $n x=a$ has a solution in $A$.

Prove that $F$ is a vector space over the field $E$.

If $F$ is finite, then $\left|F\right|=\left|E\right|^{n}$, where $n$ is the dimension of $F$ as a vector space over $E$.

If $F$ is finite, then $\left|F\right|=p^{m}$, where $p$ is the characteristic of $F$.

Find a basis and the dimension of the field $\mathbb {C}$ over the field $\mathbb {R}$.

Let $m_{1}, \ldots , m_{1}$ be distinct natural numbers, none of which is divisible by the square of a prime. Prove that the numbers $1, \sqrt{m_{1}}, \ldots , \sqrt{m_{n}}$ are linearly independent in the space $\mathbb {R}$ over $\mathbb {Q}$.

Let $r_{1}, \ldots , r_{n}$ be distinct rational numbers from the interval $(0,1)$. Prove that in the space $\mathbb {R}$ over the field $\mathbb {Q}$ the numbers $2^{r_{1}}, \ldots , 2^{r_{n}}$ are independent.

Let $\alpha$ be a complex root of a polynomial $p \in \mathbb {Q}\left[x\right]$ irreducible over $\mathbb {Q}$. Find the dimension over $\mathbb {Q}$ of the space $\mathbb {Q}\left[\alpha \right]$, consisting of numbers of the form $f(\alpha ), f \in \mathbb {Q}\left[x\right]$.

Prove that with respect to these operations the set $2^{M}$ is a vector space over the field $\mathbf{Z}_{2}$, and find a basis and its dimension.

Let $X_{1}, \ldots , X_{k}$ be subsets of $M$ such that none of them is contained in the union of the others. Prove that $X_{1}, \ldots , X_{k}$ is a linearly independent system.

$e_{1}=(1,1,1), \quad e_{2}=(1,1,2), \quad e_{3}=(1,2,3), \quad x=(6,9,14)$;

$e_{1}=(2,1,-3), \quad e_{2}=(3,2,-5), \quad e_{3}=(1,-1,1), \quad x=(6,2,-7)$;

$e_{1}=(1,2,-1,-2), \quad e_{2}=(2,3,0,-1), \quad e_{3}=(1,2,1,4)$, $e_{4}=(1,3,-1,0), \quad x=(7,14,-1,2)$. Prove that $\left(e_{1}, \ldots , e_{n}\right)$ is also a basis of the space, and find the coordinates of the vector $x$ in this basis.

$S=((1,2,1),(2,3,3),(3,8,2))$, $S^{\prime }=((3,5,8),(5,14,13),(1,9,2)) ;$

$S=((1,1,1,1),(1,2,1,1),(1,1,2,1),,(1,3,2,3))$, $S^{\prime }=((1,0,3,3),(-2,-3,-5,-4),(2,2,5,4),(-2,-3,-4,-4))$.

two vectors of the first basis are swapped;

two vectors of the second basis are swapped;

the vectors of both bases are written in reverse order?

$a_{1}=(2,2,7,-1), \quad a_{2}=(3,-1,2,4), \quad a_{3}=(1,1,3,1)$;

$a_{1}=(2,3,-4,-1), \quad a_{2}=(1,-2,1,3)$;

$a_{1}=(4,3,-1,1,1), \quad a_{2}=(2,1,-3,2,-5)$, $a_{3}=(1,-3,0,1,-2), \quad a_{4}=(1,5,2,-2,6) ;$

$a_{1}=(2,3,5,-4,1), \quad a_{2}=(1,-1,2,3,5)$.

the vectors of a plane with origin $O$ whose endpoints lie on one of two lines intersecting at the point $O$;

the vectors of a plane with origin $O$ whose endpoints lie on a given line;

the vectors of a plane with origin $O$ whose endpoints do not lie on a given line;

the vectors of the coordinate plane whose endpoints lie in the first quadrant;

the vectors of the space $\mathbb {R}^{n}$ whose coordinates are integers;

the vectors of the arithmetic space $F^{n}$, $F$ a field, that are solutions of a given system of linear equations;

the vectors of a linear space that are linear combinations of given vectors $a_{1}, \ldots , a_{k}$;

bounded sequences of complex numbers;

sequences of real numbers that have a limit;

sequences of real numbers having limit $a$;

sequences $u(n)$ of elements of the field $F$ satisfying the recurrence relation

where $(f(n))$ is a fixed sequence of elements of the field $F$, $k$ is a fixed natural number, $a_{i} \in F$;

polynomials of even degree with coefficients from the field $F$;

polynomials with coefficients from the field $F$ containing no even powers of the variable $x$;

sets from the space $2^{M}$ (see 4.9) consisting of an even number of elements;

sets from $2^{M}$ consisting of an odd number of elements.

the vectors whose first and last coordinates coincide;

the vectors whose coordinates with even indices equal 0;

the vectors whose coordinates with even indices are all equal to one another;

vectors of the form $(\alpha , \beta , \alpha , \beta , \ldots )$;

the vectors that are solutions of a homogeneous system of equations.

all matrices;

symmetric matrices;

skew-symmetric matrices;

nonsingular matrices;

singular matrices;

matrices with trace equal to zero;

matrices that commute with a given set of matrices (when computing the basis and dimension, assume that the given set of matrices consists of a single diagonal matrix with distinct diagonal entries);

matrices $X$ satisfying the condition $A_{i} X+X B_{i}=0$, where $\left\{ A_{i}, B_{i}\right\}$ is a given collection of matrices.

the functions taking the value $a$ at a given point $s \in S$;

the functions taking the value $a$ at all points of some subset $T \subseteq S$;

the functions vanishing at at least one point of $S$;

the functions having limit $a$ as $x \rightarrow \infty$ (for $S=\mathbb {R}$);

the functions having at most a finite number of discontinuities (for $S=\mathbb {R}$).

sequences in which only finitely many entries are nonzero;

sequences in which only finitely many entries equal zero;

sequences in which all entries differ from 1.

sequences satisfying the Cauchy condition: for every $\varepsilon >0$ there is a number $N \in \mathbb {N}$ such that for all $n, k>N$ the inequality $\left|x_{n}-x_{k}\right|<\varepsilon$ holds;

sequences satisfying the Hilbert condition: the series $\sum_{i=1}^{\infty }\left|x_{i}\right|^{2}$ converges;

sequences of polynomial growth, i.e., $\left|x_{n}\right| \leqslant C n^{k}$, where $C, k$ are natural numbers depending on the sequence;

sequences of exponential growth, i.e., $\left|x_{n}\right| \leqslant C e^{n}$, where $C$ is a positive real number depending on the sequence.

polynomials having a given root $\alpha \in \mathbb {R}$;

polynomials having a given root $\alpha \in \mathbb {C} \backslash \mathbb {R}$;

polynomials having given roots $\alpha_{1}, \ldots , \alpha_{k} \in \mathbb {R}$;

polynomials having a given simple root $\alpha \in \mathbb {R}$.

the dimension of the subspace of all homogeneous polynomials of degree $k$;

the dimension of its subspace consisting of all polynomials of degree $\leqslant k$.

the number of vectors in the space $V$;

the number of bases of the space $V$;

the number of nonsingular matrices of order $n$ over the field $F$;

the number of singular matrices of order $n$ over the field $F$;

the number of $k$-dimensional subspaces of the space $V$;

the number of solutions of the equation $A X=0$, where $A$ is a rectangular matrix of rank $r$, and $X$ is a column of $n$ unknowns.

$a_{1}=(1,0,0,-1), \quad a_{2}=(2,1,1,0), \quad a_{3}=(1,1,1,1)$, $a_{4}=(1,2,3,4), \quad a_{5}=(0,1,2,3) ;$

$a_{1}=(1,1,1,1,0), \quad a_{2}=(1,1,-1,-1,-1), \quad a_{3}=(2,2,0,0,-1)$, $a_{4}=(1,1,5,5,2), \quad a_{5}=(1,-1,-1,0,0)$.

if $L_{1} \subseteq L_{2}$, then $\operatorname {dim} L_{1} \leqslant \operatorname {dim} L_{2}$, with equality only when $L_{1}=L_{2}$;

if $\operatorname {dim}\left(L_{1}+L_{2}\right)=1+\operatorname {dim}\left(L_{1} \cap L_{2}\right)$, then the sum $L_{1}+L_{2}$ equals one of these subspaces, and the intersection $L_{1} \cap L_{2}$ equals the other;

if $\operatorname {dim} L_{1}+\operatorname {dim} L_{2}>\operatorname {dim} V$, then $L_{1} \cap L_{2} \neq 0$.

Is it true that $U \cap (V+W)=(U \cap V)+(U \cap W)$?

Prove that the preceding equality holds if $V \subseteq U$.

Prove that $(U+W) \cap (W+V) \cap (V+U)=[(W+V) \cap U]+[(V+U) \cap W]$.

Prove that $\operatorname {dim}[(U+V) \cap W]+\operatorname {dim}(U \cap V)=\operatorname {dim}[(V+W) \cap U]+\operatorname {dim}(V \cap W)$.

Prove that $(U \cap V)+(V \cap W)+(W \cap U) \subseteq (U+V) \cap (V+W) \cap (W+U)$ and that the difference of the dimensions of these subspaces is an even number.

$S=\langle (1,2,0,1),(1,1,1,0)\rangle$, $T=\langle (1,0,1,0),(1,3,0,1)\rangle ;$

$S=\langle (1,1,1,1),(1,-1,1,-1),(1,3,1,3)\rangle$, $T=\langle (1,2,0,2),(1,2,1,2),(3,1,3,1)\rangle ;$

$S=\langle (2,-1,0,-2),(3,-2,1,0),(1,-1,1,-1)\rangle$, $T=\langle (3,-1,-1,0),(0,-1,2,3),(5,-2,-1,0)\rangle$.

$a_{1}=(1,2,1), \quad b_{1}=(1,2,2)$, $a_{2}=(1,1,-1), \quad b_{2}=(2,3,-1)$, $a_{3}=(1,3,3), \quad b_{3}=(1,1,-3) ;$

$a_{1}=(-1,6,4,7,-2), \quad b_{1}=(1,1,2,1,-1)$, $a_{2}=(-2,3,0,5,-2), \quad b_{2}=(0,-2,0,-1,-5)$, $a_{3}=(-3,6,5,6,-5), \quad b_{3}=(2,0,2,1,-3)$;

$a_{1}=(1,1,0,0,-1), \quad b_{1}=(1,0,1,0,1)$, $a_{2}=(0,1,1,0,1), \quad b_{2}=(0,2,1,1,0)$, $a_{3}=(0,0,1,1,1), \quad b_{3}=(1,2,1,2,-1) ;$

$a_{1}=(1,2,1,0), \quad b_{1}=(2,-1,0,1)$, $a_{2}=(-1,1,1,1), \quad b_{2}=(1,-1,3,7) ;$

$a_{1}=(1,2,-1,-2), \quad b_{1}=(2,5,-6,-5)$, $a_{2}=(3,1,1,1), \quad b_{2}=(-1,2,-7,-3)$. $a_{3}=(-1,0,1,-1)$,

$\langle (1,-1,1,0),(1,1,0,1),(2,0,1,1)\rangle$;

$\langle (1,-1,1,-1,1),(1,1,0,0,3),(3,1,1,-1,7)\rangle$.

the sum of these subspaces is direct if and only if at least one of its vectors is uniquely representable in the form

the condition $L_{i} \cap L_{j}=0$ for any distinct $i$ and $j$ from 1 to $k$ is not sufficient for the sum of these subspaces to be direct.

Prove that $U \oplus V=\mathbf{M}_{n}(\mathbb {R})$.

Find the projection of the matrices $E_{i j}$ onto $U$ and $V$.

Prove that $U \oplus V=\mathbf{M}_{n}(\mathbb {R})$.

Find the projection of the matrix $E_{i j}$ onto $U$ and $V$.

the number of linear maps of $F^{n}$ into the space $F^{k}$;

the number of linear injective maps of $F^{n}$ into $F^{k}$;

the number of linear surjective maps of $F^{n}$ into $F^{k}$.

$\operatorname {Ker} \mathcal{A} \leqslant \operatorname {Ker} \mathcal{B}$;

$\mathcal{B}=\mathcal{C} \mathcal{A}$ for some operator $\mathcal{C}$ on $W$.

$\operatorname {Im} \mathcal{A} \subseteq \operatorname {Im} \mathcal{B}$;

$\mathcal{A}=\mathcal{B} \mathcal{D}$ for some operator $\mathcal{D}$ on $V$.

$\alpha^{0}, \alpha^{1}, \ldots , \alpha^{n}$;

$\beta^{0}, \beta^{1}, \ldots , \beta^{n}$;

$\gamma^{0}, \gamma^{1}, \ldots , \gamma^{n}$; are bases of the dual space $V^{*}$.

Prove that for every basis of the dual space $V^{*}$ there is a unique basis of the space $V$ for which the given basis is dual.

Find this basis in problem 36.9, a).

Find this basis in problem 36.9, b).